Discrete isothermic surfaces are defined as the maps

$F:{\mathbb{Z}}^{2}\to {\mathbb{R}}^{3}$ such that all elementary quadrilaterals of the surface have cross ratio

$-1$. 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

$F:{\mathbb{Z}}^{2}\to \u2102$ such that all elementary quadrilaterals have cross ratio

$-1$. 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.