The two main technologies for handling freeform surfaces are non-uniform rational basis-splines (NURBS) and subdivision surfaces. NURBS are the dominant standard for computer-aided design, while subdivision surfaces are popular in animation and entertainment. However there are benefits of subdivision surfaces that would be useful within computer-aided design, and features of NURBS that would make good additions to current subdivision surfaces. This thesis presents NURBS-compatible subdivision surfaces that combine topological freedom with the ability to represent any existing NURBS surface exactly. This is the first time that subdivision surfaces have been able to extend non-uniform and general-degree B-spline surfaces simultaneously. This is achieved through a novel factorisation of B-spline knot insertion rules. The thesis also shows that it is possible to bound the curvatures of the subdivision surfaces created using the author's factorisation. The resulting NURBS-compatible surface representation supports arbitrarytopology, non-uniform and general-degree surfaces, while guaranteeing high-quality second-order surface properties.