We introduce a new class of subdivision surfaces which generalize uniform tensor product B-spline surfaces of any bi-degree to meshes of arbitrary topology. Surprisingly, this can be done using subdivision rules that involve direct neighbors only. Consequently, our schemes are easy to implement, regardless of degree. The famous Catmull-Clark scheme is a special case. Similarly we show that the triangular box splines of total degree 3m + 1 can be generalized to arbitrary triangulations. Loop subdivision surfaces are a special case when m=1. our new schemes should be of interest to the high-end design market where surfaces of bi-degree up to 7 are common.