diff --git a/core/src/main/scala/cats/Foldable.scala b/core/src/main/scala/cats/Foldable.scala
index 16749333d0..5a5364170a 100644
--- a/core/src/main/scala/cats/Foldable.scala
+++ b/core/src/main/scala/cats/Foldable.scala
@@ -3,6 +3,7 @@ package cats
 import scala.collection.mutable
 import cats.instances.either._
 import cats.instances.long._
+import cats.kernel.CommutativeMonoid
 import simulacrum.typeclass
 
 /**
@@ -24,12 +25,8 @@ import simulacrum.typeclass
  *
  * See: [[http://www.cs.nott.ac.uk/~pszgmh/fold.pdf A tutorial on the universality and expressiveness of fold]]
  */
-@typeclass trait Foldable[F[_]] { self =>
+@typeclass trait Foldable[F[_]] extends UnorderedFoldable[F] { self =>
 
-  /**
-   * Left associative fold on 'F' using the function 'f'.
-   */
-  def foldLeft[A, B](fa: F[A], b: B)(f: (B, A) => B): B
 
   /**
    * Right associative lazy fold on `F` using the folding function 'f'.
@@ -301,7 +298,7 @@ import simulacrum.typeclass
   /**
    * Find the first element matching the predicate, if one exists.
    */
-  def find[A](fa: F[A])(f: A => Boolean): Option[A] =
+  override def find[A](fa: F[A])(f: A => Boolean): Option[A] =
     foldRight(fa, Now(Option.empty[A])) { (a, lb) =>
       if (f(a)) Now(Some(a)) else lb
     }.value
@@ -311,7 +308,7 @@ import simulacrum.typeclass
    *
    * If there are no elements, the result is `false`.
    */
-  def exists[A](fa: F[A])(p: A => Boolean): Boolean =
+  override def exists[A](fa: F[A])(p: A => Boolean): Boolean =
     foldRight(fa, Eval.False) { (a, lb) =>
       if (p(a)) Eval.True else lb
     }.value
@@ -321,7 +318,7 @@ import simulacrum.typeclass
    *
    * If there are no elements, the result is `true`.
    */
-  def forall[A](fa: F[A])(p: A => Boolean): Boolean =
+  override def forall[A](fa: F[A])(p: A => Boolean): Boolean =
     foldRight(fa, Eval.True) { (a, lb) =>
       if (p(a)) lb else Eval.False
     }.value
@@ -460,10 +457,10 @@ import simulacrum.typeclass
   /**
    * Returns true if there are no elements. Otherwise false.
    */
-  def isEmpty[A](fa: F[A]): Boolean =
+  override def isEmpty[A](fa: F[A]): Boolean =
     foldRight(fa, Eval.True)((_, _) => Eval.False).value
 
-  def nonEmpty[A](fa: F[A]): Boolean =
+  override def nonEmpty[A](fa: F[A]): Boolean =
     !isEmpty(fa)
 
   /**
@@ -502,6 +499,11 @@ import simulacrum.typeclass
       val F = self
       val G = Foldable[G]
     }
+
+  override def unorderedFold[A: CommutativeMonoid](fa: F[A]): A = fold(fa)
+
+  override def unorderedFoldMap[A, B: CommutativeMonoid](fa: F[A])(f: (A) => B): B =
+    foldMap(fa)(f)
 }
 
 object Foldable {
diff --git a/core/src/main/scala/cats/Traverse.scala b/core/src/main/scala/cats/Traverse.scala
index cbfbae2712..3e1f1ac5fb 100644
--- a/core/src/main/scala/cats/Traverse.scala
+++ b/core/src/main/scala/cats/Traverse.scala
@@ -16,7 +16,7 @@ import simulacrum.typeclass
  *
  * See: [[https://www.cs.ox.ac.uk/jeremy.gibbons/publications/iterator.pdf The Essence of the Iterator Pattern]]
  */
-@typeclass trait Traverse[F[_]] extends Functor[F] with Foldable[F] { self =>
+@typeclass trait Traverse[F[_]] extends Functor[F] with Foldable[F] with UnorderedTraverse[F] { self =>
 
   /**
    * Given a function which returns a G effect, thread this effect
@@ -124,4 +124,10 @@ import simulacrum.typeclass
    */
   def zipWithIndex[A](fa: F[A]): F[(A, Int)] =
     mapWithIndex(fa)((a, i) => (a, i))
+
+  override def unorderedTraverse[G[_] : CommutativeApplicative, A, B](sa: F[A])(f: (A) => G[B]): G[F[B]] =
+    traverse(sa)(f)
+
+  override def unorderedSequence[G[_] : CommutativeApplicative, A](fga: F[G[A]]): G[F[A]] =
+    sequence(fga)
 }
diff --git a/core/src/main/scala/cats/syntax/foldable.scala b/core/src/main/scala/cats/syntax/foldable.scala
index 87902c3ca3..8c686be4c8 100644
--- a/core/src/main/scala/cats/syntax/foldable.scala
+++ b/core/src/main/scala/cats/syntax/foldable.scala
@@ -1,7 +1,7 @@
 package cats
 package syntax
 
-trait FoldableSyntax extends Foldable.ToFoldableOps {
+trait FoldableSyntax extends Foldable.ToFoldableOps with UnorderedFoldable.ToUnorderedFoldableOps {
   implicit final def catsSyntaxNestedFoldable[F[_]: Foldable, G[_], A](fga: F[G[A]]): NestedFoldableOps[F, G, A] =
     new NestedFoldableOps[F, G, A](fga)
 }
diff --git a/laws/src/main/scala/cats/laws/FoldableLaws.scala b/laws/src/main/scala/cats/laws/FoldableLaws.scala
index 8adc63040b..cad1900c60 100644
--- a/laws/src/main/scala/cats/laws/FoldableLaws.scala
+++ b/laws/src/main/scala/cats/laws/FoldableLaws.scala
@@ -5,7 +5,7 @@ import cats.implicits._
 
 import scala.collection.mutable
 
-trait FoldableLaws[F[_]] {
+trait FoldableLaws[F[_]] extends UnorderedFoldableLaws[F] {
   implicit def F: Foldable[F]
 
   def leftFoldConsistentWithFoldMap[A, B](
@@ -26,12 +26,6 @@ trait FoldableLaws[F[_]] {
     fa.foldMap(f) <-> fa.foldRight(Later(M.empty))((a, lb) => lb.map(f(a) |+| _)).value
   }
 
-  def existsConsistentWithFind[A](
-    fa: F[A],
-    p: A => Boolean
-  ): Boolean = {
-    F.exists(fa)(p) == F.find(fa)(p).isDefined
-  }
 
   def existsLazy[A](fa: F[A]): Boolean = {
     var i = 0
@@ -51,31 +45,6 @@ trait FoldableLaws[F[_]] {
     i == (if (F.isEmpty(fa)) 0 else 1)
   }
 
-  def forallConsistentWithExists[A](
-    fa: F[A],
-    p: A => Boolean
-  ): Boolean = {
-    if (F.forall(fa)(p)) {
-      val negationExists = F.exists(fa)(a => !(p(a)))
-
-      // if p is true for all elements, then there cannot be an element for which
-      // it does not hold.
-      !negationExists &&
-        // if p is true for all elements, then either there must be no elements
-        // or there must exist an element for which it is true.
-        (F.isEmpty(fa) || F.exists(fa)(p))
-    } else true // can't test much in this case
-  }
-
-  /**
-   * If `F[A]` is empty, forall must return true.
-   */
-  def forallEmpty[A](
-    fa: F[A],
-    p: A => Boolean
-  ): Boolean = {
-    !F.isEmpty(fa) || F.forall(fa)(p)
-  }
 
   /**
    * Monadic folding with identity monad is analogous to `foldLeft`.
diff --git a/laws/src/main/scala/cats/laws/TraverseLaws.scala b/laws/src/main/scala/cats/laws/TraverseLaws.scala
index e7929fe4bf..3bcfc29452 100644
--- a/laws/src/main/scala/cats/laws/TraverseLaws.scala
+++ b/laws/src/main/scala/cats/laws/TraverseLaws.scala
@@ -6,7 +6,7 @@ import cats.data.{Const, Nested, State, StateT}
 import cats.syntax.traverse._
 import cats.syntax.foldable._
 
-trait TraverseLaws[F[_]] extends FunctorLaws[F] with FoldableLaws[F] {
+trait TraverseLaws[F[_]] extends FunctorLaws[F] with FoldableLaws[F] with UnorderedTraverseLaws[F] {
   implicit override def F: Traverse[F]
 
   def traverseIdentity[A, B](fa: F[A], f: A => B): IsEq[F[B]] = {
diff --git a/laws/src/main/scala/cats/laws/discipline/FoldableTests.scala b/laws/src/main/scala/cats/laws/discipline/FoldableTests.scala
index d7a3868bec..6ba4ff7fa4 100644
--- a/laws/src/main/scala/cats/laws/discipline/FoldableTests.scala
+++ b/laws/src/main/scala/cats/laws/discipline/FoldableTests.scala
@@ -2,19 +2,18 @@ package cats
 package laws
 package discipline
 
+import cats.kernel.CommutativeMonoid
 import org.scalacheck.{Arbitrary, Cogen}
 import org.scalacheck.Prop._
-import org.typelevel.discipline.Laws
-
 import cats.instances.list._
 
-trait FoldableTests[F[_]] extends Laws {
+trait FoldableTests[F[_]] extends UnorderedFoldableTests[F] {
   def laws: FoldableLaws[F]
 
   def foldable[A: Arbitrary, B: Arbitrary](implicit
     ArbFA: Arbitrary[F[A]],
-    A: Monoid[A],
-    B: Monoid[B],
+    A: CommutativeMonoid[A],
+    B: CommutativeMonoid[B],
     CogenA: Cogen[A],
     CogenB: Cogen[B],
     EqA: Eq[A],
@@ -23,12 +22,9 @@ trait FoldableTests[F[_]] extends Laws {
   ): RuleSet = {
     new DefaultRuleSet(
       name = "foldable",
-      parent = None,
+      parent = Some(unorderedFoldable[A, B]),
       "foldLeft consistent with foldMap" -> forAll(laws.leftFoldConsistentWithFoldMap[A, B] _),
       "foldRight consistent with foldMap" -> forAll(laws.rightFoldConsistentWithFoldMap[A, B] _),
-      "exists consistent with find" -> forAll(laws.existsConsistentWithFind[A] _),
-      "forall consistent with exists" -> forAll(laws.forallConsistentWithExists[A] _),
-      "forall true if empty" -> forAll(laws.forallEmpty[A] _),
       "exists is lazy" -> forAll(laws.existsLazy[A] _),
       "forall is lazy" -> forAll(laws.forallLazy[A] _),
       "foldM identity" -> forAll(laws.foldMIdentity[A, B] _),
diff --git a/laws/src/main/scala/cats/laws/discipline/NonEmptyTraverseTests.scala b/laws/src/main/scala/cats/laws/discipline/NonEmptyTraverseTests.scala
index afe6182ff3..86e2438b6c 100644
--- a/laws/src/main/scala/cats/laws/discipline/NonEmptyTraverseTests.scala
+++ b/laws/src/main/scala/cats/laws/discipline/NonEmptyTraverseTests.scala
@@ -3,28 +3,36 @@ package cats.laws.discipline
 
 import org.scalacheck.{Arbitrary, Cogen, Prop}
 import Prop.forAll
-import cats.{Applicative, Eq, Monoid, NonEmptyTraverse}
+import cats.kernel.CommutativeMonoid
+import cats.{Applicative, CommutativeApplicative, Eq, NonEmptyTraverse}
 import cats.laws.NonEmptyTraverseLaws
 
 
 trait NonEmptyTraverseTests[F[_]] extends TraverseTests[F] with ReducibleTests[F] {
   def laws: NonEmptyTraverseLaws[F]
 
-  def nonEmptyTraverse[G[_]: Applicative, A: Arbitrary, B: Arbitrary, C: Arbitrary, M: Arbitrary, X[_]: Applicative, Y[_]: Applicative](implicit
+  def nonEmptyTraverse[G[_]: Applicative, A: Arbitrary, B: Arbitrary, C: Arbitrary, M: Arbitrary, X[_], Y[_]](implicit
     ArbFA: Arbitrary[F[A]],
     ArbXB: Arbitrary[X[B]],
     ArbYB: Arbitrary[Y[B]],
     ArbYC: Arbitrary[Y[C]],
     ArbFB: Arbitrary[F[B]],
+    ArbFM: Arbitrary[F[M]],
+    ArbXM: Arbitrary[X[M]],
+    ArbYM: Arbitrary[Y[M]],
     ArbFGA: Arbitrary[F[G[A]]],
+    ArbFXM: Arbitrary[F[X[M]]],
     ArbGB: Arbitrary[G[B]],
+    ArbGM: Arbitrary[G[M]],
     CogenA: Cogen[A],
     CogenB: Cogen[B],
     CogenC: Cogen[C],
     CogenM: Cogen[M],
-    M: Monoid[M],
-    MA: Monoid[A],
-    MB: Monoid[B],
+    M: CommutativeMonoid[M],
+    MA: CommutativeMonoid[A],
+    MB: CommutativeMonoid[B],
+    CX: CommutativeApplicative[X],
+    CY: CommutativeApplicative[Y],
     EqFA: Eq[F[A]],
     EqFC: Eq[F[C]],
     EqG: Eq[G[Unit]],
@@ -33,7 +41,9 @@ trait NonEmptyTraverseTests[F[_]] extends TraverseTests[F] with ReducibleTests[F
     EqB: Eq[B],
     EqXYFC: Eq[X[Y[F[C]]]],
     EqXFB: Eq[X[F[B]]],
+    EqXFM: Eq[X[F[M]]],
     EqYFB: Eq[Y[F[B]]],
+    EqYFM: Eq[Y[F[M]]],
     EqOptionA: Eq[Option[A]]
   ): RuleSet = {
     implicit def EqXFBYFB : Eq[(X[F[B]], Y[F[B]])] = new Eq[(X[F[B]], Y[F[B]])] {
diff --git a/laws/src/main/scala/cats/laws/discipline/ReducibleTests.scala b/laws/src/main/scala/cats/laws/discipline/ReducibleTests.scala
index df9edce95b..7a0578490a 100644
--- a/laws/src/main/scala/cats/laws/discipline/ReducibleTests.scala
+++ b/laws/src/main/scala/cats/laws/discipline/ReducibleTests.scala
@@ -4,7 +4,7 @@ package discipline
 
 import cats.instances.option._
 import cats.instances.long._
-
+import cats.kernel.CommutativeMonoid
 import org.scalacheck.{Arbitrary, Cogen}
 import org.scalacheck.Prop.forAll
 
@@ -22,8 +22,8 @@ trait ReducibleTests[F[_]] extends FoldableTests[F] {
     EqA: Eq[A],
     EqB: Eq[B],
     EqOptionA: Eq[Option[A]],
-    MonoidA: Monoid[A],
-    MonoidB: Monoid[B]
+    MonoidA: CommutativeMonoid[A],
+    MonoidB: CommutativeMonoid[B]
   ): RuleSet =
     new DefaultRuleSet(
       name = "reducible",
diff --git a/laws/src/main/scala/cats/laws/discipline/TraverseTests.scala b/laws/src/main/scala/cats/laws/discipline/TraverseTests.scala
index 7cb5b1be4f..a6559cd35a 100644
--- a/laws/src/main/scala/cats/laws/discipline/TraverseTests.scala
+++ b/laws/src/main/scala/cats/laws/discipline/TraverseTests.scala
@@ -3,25 +3,29 @@ package laws
 package discipline
 
 import cats.instances.option._
-
+import cats.kernel.CommutativeMonoid
 import org.scalacheck.{Arbitrary, Cogen, Prop}
 import Prop._
 
-trait TraverseTests[F[_]] extends FunctorTests[F] with FoldableTests[F] {
+trait TraverseTests[F[_]] extends FunctorTests[F] with FoldableTests[F] with UnorderedTraverseTests[F] {
   def laws: TraverseLaws[F]
 
 
-  def traverse[A: Arbitrary, B: Arbitrary, C: Arbitrary, M: Arbitrary, X[_]: Applicative, Y[_]: Applicative](implicit
+  def traverse[A: Arbitrary, B: Arbitrary, C: Arbitrary, M: Arbitrary, X[_]: CommutativeApplicative, Y[_]: CommutativeApplicative](implicit
     ArbFA: Arbitrary[F[A]],
+    ArbFB: Arbitrary[F[B]],
     ArbXB: Arbitrary[X[B]],
+    ArbXM: Arbitrary[X[M]],
     ArbYB: Arbitrary[Y[B]],
     ArbYC: Arbitrary[Y[C]],
+    ArbYM: Arbitrary[Y[M]],
+    ArbFXM: Arbitrary[F[X[M]]],
     CogenA: Cogen[A],
     CogenB: Cogen[B],
     CogenC: Cogen[C],
     CogenM: Cogen[M],
-    M: Monoid[M],
-    MA: Monoid[A],
+    M: CommutativeMonoid[M],
+    MA: CommutativeMonoid[A],
     EqFA: Eq[F[A]],
     EqFC: Eq[F[C]],
     EqM: Eq[M],
@@ -29,6 +33,8 @@ trait TraverseTests[F[_]] extends FunctorTests[F] with FoldableTests[F] {
     EqXYFC: Eq[X[Y[F[C]]]],
     EqXFB: Eq[X[F[B]]],
     EqYFB: Eq[Y[F[B]]],
+    EqXFM: Eq[X[F[M]]],
+    EqYFM: Eq[Y[F[M]]],
     EqOptionA: Eq[Option[A]]
   ): RuleSet = {
     implicit def EqXFBYFB : Eq[(X[F[B]], Y[F[B]])] = new Eq[(X[F[B]], Y[F[B]])] {
@@ -38,7 +44,7 @@ trait TraverseTests[F[_]] extends FunctorTests[F] with FoldableTests[F] {
     new RuleSet {
       def name: String = "traverse"
       def bases: Seq[(String, RuleSet)] = Nil
-      def parents: Seq[RuleSet] = Seq(functor[A, B, C], foldable[A, M])
+      def parents: Seq[RuleSet] = Seq(functor[A, B, C], foldable[A, M], unorderedTraverse[A, M, C, X, Y])
       def props: Seq[(String, Prop)] = Seq(
         "traverse identity" -> forAll(laws.traverseIdentity[A, C] _),
         "traverse sequential composition" -> forAll(laws.traverseSequentialComposition[A, B, C, X, Y] _),
diff --git a/tests/src/test/scala/cats/tests/ListSuite.scala b/tests/src/test/scala/cats/tests/ListSuite.scala
index 8f881438f2..664426a5bb 100644
--- a/tests/src/test/scala/cats/tests/ListSuite.scala
+++ b/tests/src/test/scala/cats/tests/ListSuite.scala
@@ -16,7 +16,7 @@ class ListSuite extends CatsSuite {
   checkAll("List[Int]", AlternativeTests[List].alternative[Int, Int, Int])
   checkAll("Alternative[List]", SerializableTests.serializable(Alternative[List]))
 
-  checkAll("List[Int] with Option", TraverseTests[List].traverse[Int, Int, Int, List[Int], Option, Option])
+  checkAll("List[Int] with Option", TraverseTests[List].traverse[Int, Int, Int, Set[Int], Option, Option])
   checkAll("Traverse[List]", SerializableTests.serializable(Traverse[List]))
 
   test("nel => list => nel returns original nel")(
diff --git a/tests/src/test/scala/cats/tests/NestedSuite.scala b/tests/src/test/scala/cats/tests/NestedSuite.scala
index 8ad49c23f7..8bb3faa3ae 100644
--- a/tests/src/test/scala/cats/tests/NestedSuite.scala
+++ b/tests/src/test/scala/cats/tests/NestedSuite.scala
@@ -115,7 +115,7 @@ class NestedSuite extends CatsSuite {
   {
     // Traverse composition
     implicit val instance = ListWrapper.traverse
-    checkAll("Nested[List, ListWrapper, ?]", TraverseTests[Nested[List, ListWrapper, ?]].traverse[Int, Int, Int, List[Int], Option, Option])
+    checkAll("Nested[List, ListWrapper, ?]", TraverseTests[Nested[List, ListWrapper, ?]].traverse[Int, Int, Int, Set[Int], Option, Option])
     checkAll("Traverse[Nested[List, ListWrapper, ?]]", SerializableTests.serializable(Traverse[Nested[List, ListWrapper, ?]]))
   }
 
diff --git a/tests/src/test/scala/cats/tests/QueueSuite.scala b/tests/src/test/scala/cats/tests/QueueSuite.scala
index 621724855b..aa75a474d2 100644
--- a/tests/src/test/scala/cats/tests/QueueSuite.scala
+++ b/tests/src/test/scala/cats/tests/QueueSuite.scala
@@ -18,7 +18,7 @@ class QueueSuite extends CatsSuite {
   checkAll("Queue[Int]", MonadTests[Queue].monad[Int, Int, Int])
   checkAll("Monad[Queue]", SerializableTests.serializable(Monad[Queue]))
 
-  checkAll("Queue[Int] with Option", TraverseTests[Queue].traverse[Int, Int, Int, List[Int], Option, Option])
+  checkAll("Queue[Int] with Option", TraverseTests[Queue].traverse[Int, Int, Int, Set[Int], Option, Option])
   checkAll("Traverse[Queue]", SerializableTests.serializable(Traverse[Queue]))
 
   test("show") {
diff --git a/tests/src/test/scala/cats/tests/StreamSuite.scala b/tests/src/test/scala/cats/tests/StreamSuite.scala
index 648f114f6e..a200cd6dba 100644
--- a/tests/src/test/scala/cats/tests/StreamSuite.scala
+++ b/tests/src/test/scala/cats/tests/StreamSuite.scala
@@ -16,7 +16,7 @@ class StreamSuite extends CatsSuite {
   checkAll("Stream[Int]", MonadTests[Stream].monad[Int, Int, Int])
   checkAll("Monad[Stream]", SerializableTests.serializable(Monad[Stream]))
 
-  checkAll("Stream[Int] with Option", TraverseTests[Stream].traverse[Int, Int, Int, List[Int], Option, Option])
+  checkAll("Stream[Int] with Option", TraverseTests[Stream].traverse[Int, Int, Int, Set[Int], Option, Option])
   checkAll("Traverse[Stream]", SerializableTests.serializable(Traverse[Stream]))
 
   test("show") {
diff --git a/tests/src/test/scala/cats/tests/VectorSuite.scala b/tests/src/test/scala/cats/tests/VectorSuite.scala
index 9c9137cfd5..18aad58e86 100644
--- a/tests/src/test/scala/cats/tests/VectorSuite.scala
+++ b/tests/src/test/scala/cats/tests/VectorSuite.scala
@@ -15,7 +15,7 @@ class VectorSuite extends CatsSuite {
   checkAll("Vector[Int]", AlternativeTests[Vector].alternative[Int, Int, Int])
   checkAll("Alternative[Vector]", SerializableTests.serializable(Alternative[Vector]))
 
-  checkAll("Vector[Int] with Option", TraverseTests[Vector].traverse[Int, Int, Int, List[Int], Option, Option])
+  checkAll("Vector[Int] with Option", TraverseTests[Vector].traverse[Int, Int, Int, Set[Int], Option, Option])
   checkAll("Traverse[Vector]", SerializableTests.serializable(Traverse[Vector]))
 
   test("show") {