Discrete isothermic surfaces are defined as the maps
such that all elementary quadrilaterals of the surface have cross ratio
. It is shown that these discrete surfaces possess properties which are characteristic for smooth isothermic surfaces (Möbius invariance, dual surface). Quaternionic zero-curvature loop group representations for smooth and discrete isothermic surfaces are presented. Discrete holomorphic maps are defined as the maps
such that all elementary quadrilaterals have cross ratio
. A Weierstrass type representation for the discrete minimal isothermic surfaces (which are a special class of discrete isothermic surfaces) in terms of discrete holomorphic maps is obtained and the discrete catenoid and the Enneper surface are constructed.