  We review recent results on asymptotic lattices and their integrable
reductions. We present the theory of general asymptotic lattices in R^3
together with the corresponding theory of their Darboux-type transformations.
Then we study the discrete analogues of the Bianchi surfaces and their
transformations. Finally, we present the corresponding theory of the discrete
analogues of the isothermally-asymptotic (Fubuni-Ragazzi) nets.
