  A novel class of discrete integrable surfaces is recorded. This class of
discrete O surfaces is shown to include discrete analogues of classical
surfaces such as isothermic, `linear' Weingarten, Guichard and Petot surfaces.
Moreover, natural discrete analogues of the Gaussian and mean curvatures for
surfaces parametrized in terms of curvature coordinates are used to define
surfaces of constant discrete Gaussian and mean curvatures and discrete minimal
surfaces. Remarkably, these turn out to be proto-typical examples of discrete O
surfaces. It is demonstrated that the construction of a Backlund transformation
for discrete O surfaces leads in a natural manner to an associated
parameter-dependent linear representation. Canonical discretizations of the
classical pseudosphere and breather pseudospherical surfaces are generated.
Connections with pioneering work by Bobenko and Pinkall are established.
