diff --git a/src/lib.rs b/src/lib.rs
index 3b3f3671c..2a618b780 100644
--- a/src/lib.rs
+++ b/src/lib.rs
@@ -25,3 +25,5 @@ pub mod values;
 
 pub use crate::extension::Extension;
 pub use crate::hugr::{Direction, Hugr, HugrView, Node, Port, SimpleReplacement, Wire};
+
+pub mod walker;
diff --git a/src/walker.rs b/src/walker.rs
new file mode 100644
index 000000000..c06ed0bd1
--- /dev/null
+++ b/src/walker.rs
@@ -0,0 +1,310 @@
+#![allow(missing_docs)]
+
+use std::collections::HashSet;
+
+use derive_more::{Deref, DerefMut};
+use itertools::Itertools;
+
+use crate::{ops::OpType, HugrView, Node};
+
+#[derive(Eq, PartialEq, Ord, PartialOrd, Copy, Clone, Debug)]
+pub enum WalkOrder {
+    Preorder,
+    Postorder,
+}
+
+#[derive(Deref, DerefMut)]
+struct WalkerCallback<'a, T, E>(Box<dyn 'a + FnMut(Node, T) -> Result<T, E>>);
+
+impl<'a, T, E, F: 'a + FnMut(Node, T) -> Result<T, E>> From<F> for WalkerCallback<'a, T, E> {
+    fn from(f: F) -> Self {
+        Self(Box::new(f))
+    }
+}
+
+pub struct Walker<'a, H: HugrView, T, E> {
+    pre_callbacks: Vec<WalkerCallback<'a, T, E>>,
+    post_callbacks: Vec<WalkerCallback<'a, T, E>>,
+    hugr: &'a H,
+}
+
+impl<'a, H: HugrView, T, E> Walker<'a, H, T, E> {
+    pub fn new(hugr: &'a H) -> Self {
+        Self {
+            pre_callbacks: Vec::new(),
+            post_callbacks: Vec::new(),
+            hugr,
+        }
+    }
+
+    pub fn previsit<O, F: 'a + FnMut(Node, O, T) -> Result<T, E>>(&mut self, f: F) -> &mut Self
+    where
+        OpType: TryInto<O>,
+    {
+        self.visit(WalkOrder::Preorder, f)
+    }
+
+    pub fn postvisit<O, F: 'a + FnMut(Node, O, T) -> Result<T, E>>(&mut self, f: F) -> &mut Self
+    where
+        OpType: TryInto<O>,
+    {
+        self.visit(WalkOrder::Postorder, f)
+    }
+
+    fn mut_callbacks(&mut self, order: WalkOrder) -> &mut Vec<WalkerCallback<'a, T, E>> {
+        match order {
+            WalkOrder::Preorder => &mut self.pre_callbacks,
+            WalkOrder::Postorder => &mut self.post_callbacks,
+        }
+    }
+
+    pub fn visit<O, F: 'a + FnMut(Node, O, T) -> Result<T, E>>(
+        &mut self,
+        walk_order: WalkOrder,
+        mut f: F,
+    ) -> &mut Self
+    where
+        OpType: TryInto<O>,
+    {
+        let hugr = self.hugr;
+        let cb = move |n, t| match hugr.get_optype(n).clone().try_into() {
+            Ok(x) => f(n, x, t),
+            _ => Ok(t),
+        };
+        self.mut_callbacks(walk_order).push(cb.into());
+        self
+    }
+
+    pub fn walk(&mut self, mut t: T) -> Result<T, E> {
+        enum WorkItem {
+            Visit(Node),
+            Callback(WalkOrder, Node),
+        }
+        impl From<Node> for WorkItem {
+            fn from(n: Node) -> Self {
+                WorkItem::Visit(n)
+            }
+        }
+        // We intentionally avoid recursion so that we can robustly accept very deep hugrs
+        let mut worklist = vec![self.hugr.root().into()];
+
+        while let Some(wi) = worklist.pop() {
+            match wi {
+                WorkItem::Visit(n) => {
+                    worklist.push(WorkItem::Callback(WalkOrder::Postorder, n));
+                    let mut pushed_children = HashSet::new();
+                    // We intend to only visit direct children.
+                    //
+                    // If the nodes children form a dataflow sibling graph we
+                    // visit them in post dfs order starting from the Input
+                    // node. Then (whether or not it's a dataflow sibling graph)
+                    // we visit each remaining unvisited child in children() order.
+                    //
+                    // The second traversal is required to ensure we visit both
+                    // nodes unreachable from Input in a dataflow sibling graph
+                    // (e.g. LoadConstant) and the children of non dataflow
+                    // sibling graph nodes (e.g. the children of CFG or Conditional
+                    // nodes)
+                    if let Some([input, _]) = self.hugr.get_io(n) {
+                        let petgraph = self.hugr.as_petgraph();
+                        // Here we visit the nodes in DfsPostOrder(i.e. we have
+                        // visited all the out_neighbors() of a node before we
+                        // visit that node), and push a node onto the worklist
+                        // stack when we visit it. So once we are done the stack
+                        // will have the Input node at the top, and a nodes
+                        // out_neighbors are always under that node on the
+                        // worklist stack.
+                        let mut dfs = ::petgraph::visit::DfsPostOrder::new(&petgraph, input);
+                        while let Some(x) = dfs.next(&petgraph) {
+                            worklist.push(x.into());
+                            pushed_children.insert(x);
+                        }
+                    }
+
+                    // Here we collect all children that were not visited by the
+                    // DfsPostOrder traversal above, in children() order
+                    let rest_children = self
+                        .hugr
+                        .children(n)
+                        .filter(|x| !pushed_children.contains(x))
+                        .collect_vec();
+                    // We extend in reverse so that the first child is the top of the stack
+                    worklist.extend(rest_children.into_iter().rev().map(WorkItem::Visit));
+                    worklist.push(WorkItem::Callback(WalkOrder::Preorder, n));
+                }
+                WorkItem::Callback(order, n) => {
+                    for cb in self.mut_callbacks(order).iter_mut() {
+                        t = cb(n, t)?;
+                    }
+                }
+            }
+        }
+        Ok(t)
+    }
+}
+
+/// An example of use using the Walker to implement an iterator over all nodes,
+/// nodes are visited in preorder where possible. More precisely, nodes are
+/// visited before their children, and nodes in a dataflow sibling graph are
+/// visited before their out_neighbours.
+pub fn hugr_walk_iter(h: &impl HugrView) -> impl Iterator<Item = Node> {
+    let mut walker = Walker::<_, Vec<Node>, std::convert::Infallible>::new(h);
+    walker.previsit(|n, _: OpType, mut v| {
+        v.push(n);
+        Ok(v)
+    });
+    walker.walk(Vec::new()).unwrap().into_iter()
+}
+
+/// An example of use using the Walker to implement a search.
+/// This demonstrates terminating a walk early.
+pub fn hugr_walk_find<O, V>(h: &impl HugrView, mut f: impl FnMut(Node, O) -> Option<V>) -> Option<V>
+where
+    OpType: TryInto<O>,
+{
+    Walker::new(h)
+        .previsit(|n, o: O, ()| f(n, o).map_or(Ok(()), Result::Err))
+        .walk(())
+        .map_or_else(Option::Some, |()| None)
+}
+
+#[cfg(test)]
+mod test {
+    use std::error::Error;
+
+    use crate::builder::{Dataflow, DataflowHugr};
+    use crate::extension::prelude::USIZE_T;
+    use crate::hugr::hugrmut::sealed::HugrMutInternals;
+    use crate::ops;
+    use crate::types::Signature;
+    use crate::{
+        builder::{Container, FunctionBuilder, HugrBuilder, ModuleBuilder, SubContainer},
+        extension::{ExtensionRegistry, ExtensionSet},
+        ops::{FuncDefn, Module},
+        type_row,
+        types::FunctionType,
+    };
+
+    use super::*;
+
+    #[test]
+    fn test1() -> Result<(), Box<dyn Error>> {
+        let mut module_builder = ModuleBuilder::new();
+        let sig = Signature {
+            signature: FunctionType::new(type_row![], type_row![]),
+            input_extensions: ExtensionSet::new(),
+        };
+        module_builder
+            .define_function("f1", sig.clone())?
+            .finish_sub_container()?;
+        module_builder
+            .define_function("f2", sig.clone())?
+            .finish_sub_container()?;
+
+        let hugr = module_builder.finish_hugr(&ExtensionRegistry::new())?;
+
+        let s = Walker::<_, _, Box<dyn Error>>::new(&hugr)
+            .visit(WalkOrder::Preorder, |_, Module, mut r| {
+                r += ";prem";
+                Ok(r)
+            })
+            .visit(WalkOrder::Postorder, |_, Module, mut r| {
+                r += ";postm";
+                Ok(r)
+            })
+            .visit(
+                WalkOrder::Preorder,
+                |_, FuncDefn { ref name, .. }, mut r| {
+                    r += ";pre";
+                    r += name.as_ref();
+                    Ok(r)
+                },
+            )
+            .visit(
+                WalkOrder::Postorder,
+                |_, FuncDefn { ref name, .. }, mut r| {
+                    r += ";post";
+                    r += name.as_ref();
+                    Ok(r)
+                },
+            )
+            .walk(String::new())?;
+
+        assert_eq!(s, ";prem;pref1;postf1;pref2;postf2;postm");
+        Ok(())
+    }
+
+    struct Noop;
+
+    impl TryFrom<ops::OpType> for Noop {
+        type Error = ops::OpType;
+        fn try_from(ot: ops::OpType) -> Result<Self, Self::Error> {
+            match ot {
+                ops::OpType::LeafOp(ops::LeafOp::Noop { .. }) => Ok(Noop),
+                x => Err(x),
+            }
+        }
+    }
+    #[test]
+    fn test2() -> Result<(), Box<dyn Error>> {
+        use ops::handle::NodeHandle;
+        let sig = Signature {
+            signature: FunctionType::new(type_row![USIZE_T], type_row![USIZE_T]),
+            input_extensions: ExtensionSet::new(),
+        };
+        let mut fun_builder = FunctionBuilder::new("f3", sig)?;
+        let [i] = fun_builder.input_wires_arr();
+        let noop1 = fun_builder.add_dataflow_op(ops::LeafOp::Noop { ty: USIZE_T }, [i])?;
+        let noop2 =
+            fun_builder.add_dataflow_op(ops::LeafOp::Noop { ty: USIZE_T }, [noop1.out_wire(0)])?;
+        let mut h = fun_builder.finish_prelude_hugr_with_outputs([noop2.out_wire(0)])?;
+        h.hugr_mut()
+            .move_before_sibling(noop2.handle().node(), noop1.handle().node())?;
+
+        let v = Walker::<_, Vec<Node>, Box<dyn Error>>::new(&h)
+            .previsit(|n, Noop, mut v| {
+                v.push(n);
+                Ok(v)
+            })
+            .walk(Vec::new())?;
+        assert_eq!(
+            &[noop1.handle().node(), noop2.handle().node()],
+            v.as_slice()
+        );
+        Ok(())
+    }
+
+    #[test]
+    fn leaf_op_out_degree() {
+        use std::collections::HashMap;
+        let h: crate::Hugr = todo!();
+        let mut walker = Walker::new(&h);
+        walker.postvisit(|n, _: crate::ops::LeafOp, mut r| {
+            r.insert(n, h.node_outputs(n).map(|o| h.linked_ports(n, o).count()));
+            Ok(r)
+        });
+        let r = walker.walk(HashMap::new()).unwrap();
+        // TODO construct example and assert result of walk
+    }
+
+    #[test]
+    fn pretty_printer() {
+        struct PPCtx(usize, String);
+        let h: crate::Hugr = todo!();
+        let pp_out = Walker::<_, _, std::convert::Infallible>::new(&h)
+            .previsit(|n, _: OpType, PPCtx(mut indent, mut r)| {
+                use crate::hugr::NodeIndex;
+                r += format!(
+                    "{}{}\n",
+                    std::iter::repeat(' ').take(indent).collect::<String>(),
+                    n.index()
+                )
+                .as_str();
+                Ok(PPCtx(indent + 2, r))
+            })
+            .postvisit(|_, _: OpType, PPCtx(mut indent, r)| Ok(PPCtx(indent - 2, r)))
+            .walk(PPCtx(0, "".to_string()))
+            .unwrap();
+        // TODO construct example and assert result of walk
+    }
+}