diff --git a/src/circuit.rs b/src/circuit.rs
new file mode 100644
index 0000000..fb74fad
--- /dev/null
+++ b/src/circuit.rs
@@ -0,0 +1,143 @@
+//! Traits and structs for implementing circuit components.
+
+use std::fmt;
+
+use crate::{
+    arithmetic::FieldExt,
+    plonk::{Advice, Any, Column, ConstraintSystem, Error, Fixed},
+};
+
+pub mod layouter;
+
+/// A chip implements a set of instructions that can be used by gadgets.
+///
+/// The chip itself should not store any state; instead, state that is required at circuit
+/// synthesis time should be stored in [`Chip::Config`], which can then be fetched via
+/// [`Layouter::config`].
+pub trait Chip: Sized {
+    /// A type that holds the configuration for this chip, and any other state it may need
+    /// during circuit synthesis.
+    type Config: fmt::Debug;
+
+    /// The field that the chip is defined over.
+    ///
+    /// This provides a type that the chip's configuration can reference if necessary.
+    type Field: FieldExt;
+
+    /// Load any fixed configuration for this chip into the circuit.
+    fn load(layouter: &mut impl Layouter<Self>) -> Result<(), Error>;
+}
+
+/// A pointer to a cell within a circuit.
+#[derive(Clone, Copy, Debug)]
+pub struct Cell {
+    /// Identifies the region in which this cell resides.
+    region_index: usize,
+    row_offset: usize,
+    column: Column<Any>,
+}
+
+/// A permutation configured by a chip.
+#[derive(Clone, Debug)]
+pub struct Permutation {
+    index: usize,
+    mapping: Vec<Column<Any>>,
+}
+
+impl Permutation {
+    /// Configures a new permutation for the given columns.
+    pub fn new<F: FieldExt>(meta: &mut ConstraintSystem<F>, columns: &[Column<Advice>]) -> Self {
+        let index = meta.permutation(columns);
+        Permutation {
+            index,
+            mapping: columns.iter().map(|c| (*c).into()).collect(),
+        }
+    }
+}
+
+/// A region of the circuit in which a [`Chip`] can assign cells.
+///
+/// Inside a region, the chip may freely use relative offsets; the [`Layouter`] will
+/// treat these assignments as a single "region" within the circuit.
+///
+/// The [`Layouter`] is allowed to optimise between regions as it sees fit. Chips must use
+/// [`Region::constrain_equal`] to copy in variables assigned in other regions.
+///
+/// TODO: It would be great if we could constrain the columns in these types to be
+/// "logical" columns that are guaranteed to correspond to the chip (and have come from
+/// `Chip::Config`).
+#[derive(Debug)]
+pub struct Region<'r, C: Chip> {
+    region: &'r mut dyn layouter::RegionLayouter<C>,
+}
+
+impl<'r, C: Chip> From<&'r mut dyn layouter::RegionLayouter<C>> for Region<'r, C> {
+    fn from(region: &'r mut dyn layouter::RegionLayouter<C>) -> Self {
+        Region { region }
+    }
+}
+
+impl<'r, C: Chip> Region<'r, C> {
+    /// Assign an advice column value (witness).
+    ///
+    /// Even though `to` has `FnMut` bounds, it is guaranteed to be called at most once.
+    pub fn assign_advice<'v>(
+        &'v mut self,
+        column: Column<Advice>,
+        offset: usize,
+        mut to: impl FnMut() -> Result<C::Field, Error> + 'v,
+    ) -> Result<Cell, Error> {
+        self.region.assign_advice(column, offset, &mut to)
+    }
+
+    /// Assign a fixed value.
+    ///
+    /// Even though `to` has `FnMut` bounds, it is guaranteed to be called at most once.
+    pub fn assign_fixed<'v>(
+        &'v mut self,
+        column: Column<Fixed>,
+        offset: usize,
+        mut to: impl FnMut() -> Result<C::Field, Error> + 'v,
+    ) -> Result<Cell, Error> {
+        self.region.assign_fixed(column, offset, &mut to)
+    }
+
+    /// Constraint two cells to have the same value.
+    ///
+    /// Returns an error if either of the cells is not within the given permutation.
+    pub fn constrain_equal(
+        &mut self,
+        permutation: &Permutation,
+        left: Cell,
+        right: Cell,
+    ) -> Result<(), Error> {
+        self.region.constrain_equal(permutation, left, right)
+    }
+}
+
+/// A layout strategy for a specific chip within a circuit.
+///
+/// This abstracts over the circuit assignments, handling row indices etc.
+///
+/// A particular concrete layout strategy will implement this trait for each chip it
+/// supports.
+pub trait Layouter<C: Chip> {
+    /// Provides access to the chip configuration.
+    fn config(&self) -> &C::Config;
+
+    /// Assign a region of gates to an absolute row number.
+    ///
+    /// Inside the closure, the chip may freely use relative offsets; the `Layouter` will
+    /// treat these assignments as a single "region" within the circuit. Outside this
+    /// closure, the `Layouter` is allowed to optimise as it sees fit.
+    ///
+    /// ```ignore
+    /// fn assign_region(&mut self, |region| {
+    ///     region.assign_advice(self.config.a, offset, || { Some(value)});
+    /// });
+    /// ```
+    fn assign_region(
+        &mut self,
+        assignment: impl FnOnce(Region<'_, C>) -> Result<(), Error>,
+    ) -> Result<(), Error>;
+}
diff --git a/src/circuit/layouter.rs b/src/circuit/layouter.rs
new file mode 100644
index 0000000..fa329f5
--- /dev/null
+++ b/src/circuit/layouter.rs
@@ -0,0 +1,218 @@
+//! Implementations of common circuit layouters.
+
+use std::cmp;
+use std::fmt;
+use std::marker::PhantomData;
+
+use super::{Cell, Chip, Layouter, Permutation, Region};
+use crate::plonk::{Advice, Assignment, Column, Error, Fixed};
+
+/// Helper trait for implementing a custom [`Layouter`].
+///
+/// This trait is used for implementing region assignments:
+///
+/// ```ignore
+/// impl<'a, C: Chip, CS: Assignment<C::Field> + 'a> Layouter<C> for MyLayouter<'a, C, CS> {
+///     fn assign_region(
+///         &mut self,
+///         assignment: impl FnOnce(Region<'_, C>) -> Result<(), Error>,
+///     ) -> Result<(), Error> {
+///         let region_index = self.regions.len();
+///         self.regions.push(self.current_gate);
+///
+///         let mut region = MyRegion::new(self, region_index);
+///         {
+///             let region: &mut dyn RegionLayouter<C> = &mut region;
+///             assignment(region.into())?;
+///         }
+///         self.current_gate += region.row_count;
+///
+///         Ok(())
+///     }
+/// }
+/// ```
+///
+/// TODO: It would be great if we could constrain the columns in these types to be
+/// "logical" columns that are guaranteed to correspond to the chip (and have come from
+/// `Chip::Config`).
+pub trait RegionLayouter<C: Chip>: fmt::Debug {
+    /// Assign an advice column value (witness)
+    fn assign_advice<'v>(
+        &'v mut self,
+        column: Column<Advice>,
+        offset: usize,
+        to: &'v mut (dyn FnMut() -> Result<C::Field, Error> + 'v),
+    ) -> Result<Cell, Error>;
+
+    /// Assign a fixed value
+    fn assign_fixed<'v>(
+        &'v mut self,
+        column: Column<Fixed>,
+        offset: usize,
+        to: &'v mut (dyn FnMut() -> Result<C::Field, Error> + 'v),
+    ) -> Result<Cell, Error>;
+
+    /// Constraint two cells to have the same value.
+    ///
+    /// Returns an error if either of the cells is not within the given permutation.
+    fn constrain_equal(
+        &mut self,
+        permutation: &Permutation,
+        left: Cell,
+        right: Cell,
+    ) -> Result<(), Error>;
+}
+
+/// A [`Layouter`] for a single-chip circuit.
+pub struct SingleChip<'a, C: Chip, CS: Assignment<C::Field> + 'a> {
+    cs: &'a mut CS,
+    config: C::Config,
+    regions: Vec<usize>,
+    current_gate: usize,
+    _marker: PhantomData<C>,
+}
+
+impl<'a, C: Chip, CS: Assignment<C::Field> + 'a> fmt::Debug for SingleChip<'a, C, CS> {
+    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
+        f.debug_struct("SingleChip")
+            .field("config", &self.config)
+            .field("regions", &self.regions)
+            .field("current_gate", &self.current_gate)
+            .finish()
+    }
+}
+
+impl<'a, C: Chip, CS: Assignment<C::Field>> SingleChip<'a, C, CS> {
+    /// Creates a new single-chip layouter.
+    pub fn new(cs: &'a mut CS, config: C::Config) -> Self {
+        SingleChip {
+            cs,
+            config,
+            regions: vec![],
+            current_gate: 0,
+            _marker: PhantomData,
+        }
+    }
+}
+
+impl<'a, C: Chip, CS: Assignment<C::Field> + 'a> Layouter<C> for SingleChip<'a, C, CS> {
+    fn config(&self) -> &C::Config {
+        &self.config
+    }
+
+    fn assign_region(
+        &mut self,
+        assignment: impl FnOnce(Region<'_, C>) -> Result<(), Error>,
+    ) -> Result<(), Error> {
+        let region_index = self.regions.len();
+        self.regions.push(self.current_gate);
+
+        let mut region = SingleChipRegion::new(self, region_index);
+        {
+            let region: &mut dyn RegionLayouter<C> = &mut region;
+            assignment(region.into())?;
+        }
+        self.current_gate += region.row_count;
+
+        Ok(())
+    }
+}
+
+struct SingleChipRegion<'r, 'a, C: Chip, CS: Assignment<C::Field> + 'a> {
+    layouter: &'r mut SingleChip<'a, C, CS>,
+    region_index: usize,
+    row_count: usize,
+}
+
+impl<'r, 'a, C: Chip, CS: Assignment<C::Field> + 'a> fmt::Debug
+    for SingleChipRegion<'r, 'a, C, CS>
+{
+    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
+        f.debug_struct("SingleChipRegion")
+            .field("layouter", &self.layouter)
+            .field("region_index", &self.region_index)
+            .field("row_count", &self.row_count)
+            .finish()
+    }
+}
+
+impl<'r, 'a, C: Chip, CS: Assignment<C::Field> + 'a> SingleChipRegion<'r, 'a, C, CS> {
+    fn new(layouter: &'r mut SingleChip<'a, C, CS>, region_index: usize) -> Self {
+        SingleChipRegion {
+            layouter,
+            region_index,
+            row_count: 0,
+        }
+    }
+}
+
+impl<'r, 'a, C: Chip, CS: Assignment<C::Field> + 'a> RegionLayouter<C>
+    for SingleChipRegion<'r, 'a, C, CS>
+{
+    fn assign_advice<'v>(
+        &'v mut self,
+        column: Column<Advice>,
+        offset: usize,
+        to: &'v mut (dyn FnMut() -> Result<C::Field, Error> + 'v),
+    ) -> Result<Cell, Error> {
+        self.layouter.cs.assign_advice(
+            column,
+            self.layouter.regions[self.region_index] + offset,
+            to,
+        )?;
+        self.row_count = cmp::max(self.row_count, offset);
+
+        Ok(Cell {
+            region_index: self.region_index,
+            row_offset: offset,
+            column: column.into(),
+        })
+    }
+
+    fn assign_fixed<'v>(
+        &'v mut self,
+        column: Column<Fixed>,
+        offset: usize,
+        to: &'v mut (dyn FnMut() -> Result<C::Field, Error> + 'v),
+    ) -> Result<Cell, Error> {
+        self.layouter.cs.assign_fixed(
+            column,
+            self.layouter.regions[self.region_index] + offset,
+            to,
+        )?;
+        self.row_count = cmp::max(self.row_count, offset);
+        Ok(Cell {
+            region_index: self.region_index,
+            row_offset: offset,
+            column: column.into(),
+        })
+    }
+
+    fn constrain_equal(
+        &mut self,
+        permutation: &Permutation,
+        left: Cell,
+        right: Cell,
+    ) -> Result<(), Error> {
+        let left_column = permutation
+            .mapping
+            .iter()
+            .position(|c| c == &left.column)
+            .ok_or(Error::SynthesisError)?;
+        let right_column = permutation
+            .mapping
+            .iter()
+            .position(|c| c == &right.column)
+            .ok_or(Error::SynthesisError)?;
+
+        self.layouter.cs.copy(
+            permutation.index,
+            left_column,
+            self.layouter.regions[left.region_index] + left.row_offset,
+            right_column,
+            self.layouter.regions[right.region_index] + right.row_offset,
+        )?;
+
+        Ok(())
+    }
+}
diff --git a/src/gadget.rs b/src/gadget.rs
new file mode 100644
index 0000000..a876ca7
--- /dev/null
+++ b/src/gadget.rs
@@ -0,0 +1 @@
+//! Self-contained circuit implementations of various primitives.
diff --git a/src/lib.rs b/src/lib.rs
index ac043c4..2accaaa 100644
--- a/src/lib.rs
+++ b/src/lib.rs
@@ -14,6 +14,8 @@
 #![deny(unsafe_code)]
 
 pub mod arithmetic;
+pub mod circuit;
+pub mod gadget;
 pub mod pasta;
 pub mod plonk;
 pub mod poly;