We present an interpolatory subdivision scheme to generate adaptiely refined quadrilateral meshes which approximate a smooth surface of arbitrary topology. The described method significantly differs from classical mesh generation techniques based on spline surfaces or implicit representations since no explicit description of the limit surface is used. Instead, simple affine combinations are applied to compute new vertices if a face of the net is split. These rules are designed to guarantee asymptotic smoothness, i.e., the sequence of refined nets converges to a smooth limit surface. Subdivision techniques are useful mainly in applications where a given quadrilateral net is a coarse approximation of a surface and points on a refined grid have to be estimated. To evaluate our approach, we show examples for FE-computations on surfaces generated by this algorithm.