  We investigate the common underlying discrete structures for various smooth
and discrete nets. The main idea is to impose the characteristic properties of
the nets not only on elementary quadrilaterals but also on larger parameter
rectangles. For discrete planar quadrilateral nets, circular nets, $Q^*$-nets
and conical nets we obtain a characterization of the corresponding discrete
multi-nets. In the limit these discrete nets lead to some classical classes of
smooth surfaces. Furthermore, we propose to use the characterized discrete nets
as discrete extensions for the nets to obtain structure preserving subdivision
schemes.
