The paper deals with the investigation of triangular -subdivision schemes in the stationary shift-invariant setting. In Section 2 we collect the available theory on refinable functions (subdivision surfaces), with emphasis on their Sobolev and Hölder smoothness. Families of interpolatory and approximating -subdivision schemes are investigated in Section 3. Some dual -subdivision schemes which are related to vector-valued refinable functions are also analyzed. For this purpose, we have developed Matlab routines for numerically investigating properties of vector subdivision schemes.