  We discuss integrable discretizations of 3-dimensional cyclic systems, that
is, orthogonal coordinate systems with one family of circular coordinate lines.
In particular, the underlying circle congruences are investigated in detail,
and characterized by the existence of a certain flat connection. Within the
developed framework, discrete cyclic systems with a family of discrete flat
fronts in hyperbolic space and discrete cyclic systems, where all coordinate
surfaces are discrete Dupin cyclides, are investigated.
