In this paper we introduce a new subdivision operator that unifies triangular and quadrilateral subdivision schemes. Designers often want the added flexibility of having both quads and triangles in their models. It is also well known that triangle meshes generate poor limit surfaces when using a quad scheme, while quad-only meshes behave poorly with triangular schemes. Our new scheme is a generalization of the well known Catmull-Clark and Loop subdivision algorithms. We show that our surfaces are C1 everywhere and provide a proof that it is impossible to construct a C2 scheme at the quad/triangle boundary. However, we provide rules that produce surfaces with bounded curvature at the regular quad/triangle boundary and provide optimal masks that minimize the curvature divergence elsewhere. We demonstrate the visual quality of our surfaces with several examples.