Asymptotic lattices and their integrable reductions I: the Bianchi and the Fubini-Ragazzi lattices

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.