We present the necessary theory for the integration of subdivision surfaces into general purpose rendering systems. The most important functionality that has to be provided via an abstract geometry interface are the computation of surface points and normals as well as the ray intersection test. We demonstrate how to derive the corresponding formulas and how to construct tight bounding volumes for subdivision surfaces. We introduce envelope meshes which have the same topology as the control meshes but tightly circumscribe the limit surface. An efficient and simple algorithm is presented to trace a ray recursively through the forest of triangles emerging from adaptive refinement of an envelope mesh.