diff --git a/crates/fj-kernel/src/algorithms/intersection/face_face.rs b/crates/fj-kernel/src/algorithms/intersection/face_face.rs
new file mode 100644
index 000000000..4603c6659
--- /dev/null
+++ b/crates/fj-kernel/src/algorithms/intersection/face_face.rs
@@ -0,0 +1,116 @@
+use crate::objects::{Curve, Face};
+
+use super::{CurveFaceIntersection, SurfaceSurfaceIntersection};
+
+/// An intersection between two faces
+#[derive(Clone, Debug, Eq, PartialEq, Hash, Ord, PartialOrd)]
+pub struct FaceFaceIntersection {
+    /// The intersection curves
+    ///
+    /// These curves correspond to the input faces, each being the local
+    /// representation of the intersection on the respective face's surface.
+    ///
+    /// They both represent the same global curve.
+    pub intersection_curves: [Curve; 2],
+
+    /// The interval of this intersection, in curve coordinates
+    ///
+    /// These curve coordinates apply to both intersection curves equally.
+    pub intersection_intervals: CurveFaceIntersection,
+}
+
+impl FaceFaceIntersection {
+    /// Compute the intersections between two faces
+    pub fn compute(faces: [&Face; 2]) -> Option<Self> {
+        let surfaces = faces.map(|face| face.surface());
+
+        let intersection_curves =
+            SurfaceSurfaceIntersection::compute(surfaces)?.intersection_curves;
+
+        // Can be cleaned up, once `zip` is stable:
+        // https://doc.rust-lang.org/std/primitive.array.html#method.zip
+        let curve_face_intersections = {
+            let [curve_a, curve_b] = intersection_curves;
+            let [face_a, face_b] = faces;
+
+            [(curve_a, face_a), (curve_b, face_b)].map(|(curve, face)| {
+                CurveFaceIntersection::compute(&curve, face)
+            })
+        };
+
+        let intersection_intervals = {
+            let [a, b] = curve_face_intersections;
+            a.merge(&b)
+        };
+
+        if intersection_intervals.is_empty() {
+            return None;
+        }
+
+        Some(Self {
+            intersection_curves,
+            intersection_intervals,
+        })
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use pretty_assertions::assert_eq;
+
+    use crate::{
+        algorithms::intersection::CurveFaceIntersection,
+        objects::{Curve, Face, Surface},
+    };
+
+    use super::FaceFaceIntersection;
+
+    #[test]
+    fn compute_no_intersection() {
+        #[rustfmt::skip]
+        let points = [
+            [1., 1.],
+            [2., 1.],
+            [2., 2.],
+            [1., 2.],
+        ];
+        let surfaces = [Surface::xy_plane(), Surface::xz_plane()];
+        let [a, b] = surfaces.map(|surface| {
+            Face::build(surface).polygon_from_points(points).into_face()
+        });
+
+        let intersection = FaceFaceIntersection::compute([&a, &b]);
+
+        assert!(intersection.is_none());
+    }
+
+    #[test]
+    fn compute_one_intersection() {
+        #[rustfmt::skip]
+        let points = [
+            [-1., -1.],
+            [ 1., -1.],
+            [ 1.,  1.],
+            [-1.,  1.],
+        ];
+        let surfaces = [Surface::xy_plane(), Surface::xz_plane()];
+        let [a, b] = surfaces.map(|surface| {
+            Face::build(surface).polygon_from_points(points).into_face()
+        });
+
+        let intersection = FaceFaceIntersection::compute([&a, &b]);
+
+        let expected_curves = surfaces.map(|surface| {
+            Curve::build(surface).line_from_points([[0., 0.], [1., 0.]])
+        });
+        let expected_intervals =
+            CurveFaceIntersection::from_intervals([[[-1.], [1.]]]);
+        assert_eq!(
+            intersection,
+            Some(FaceFaceIntersection {
+                intersection_curves: expected_curves,
+                intersection_intervals: expected_intervals
+            })
+        );
+    }
+}
diff --git a/crates/fj-kernel/src/algorithms/intersection/mod.rs b/crates/fj-kernel/src/algorithms/intersection/mod.rs
index 06baac98b..074e672a4 100644
--- a/crates/fj-kernel/src/algorithms/intersection/mod.rs
+++ b/crates/fj-kernel/src/algorithms/intersection/mod.rs
@@ -2,12 +2,14 @@
 
 mod curve_edge;
 mod curve_face;
+mod face_face;
 mod line_segment;
 mod surface_surface;
 
 pub use self::{
     curve_edge::CurveEdgeIntersection,
     curve_face::{CurveFaceIntersection, CurveFaceIntersectionInterval},
+    face_face::FaceFaceIntersection,
     line_segment::LineSegmentIntersection,
     surface_surface::SurfaceSurfaceIntersection,
 };