Abstract: We describe an extension of B-splines to surfaces of arbitrary topology, including arbitrary boundaries. The technique inherits many of the properties of B-splines: local control, a compact representation, and guaranteed continuity of arbitrary degree. The surface is specified using a polyhedral control mesh instead of a rectangular one; the resulting surface approximates the polyhedral mesh much as a B-spline approximates its rectangular control mesh. Like a B-spline, the surface is a single, continuous object. This is achieved by modeling the domain of the surface with a manifold whose topology matches that of the polyhedral mesh, then embedding this domain into 3-space using a basis-function/control-point formulation. We provide a constructive approach to building a manifold.