Equations that are known now as the "two-dimensional Toda lattice" have in fact appeared in classical differential geometry in the end of 19th century. Generalized 2D-Toda lattices corresponding to the Cartan matrices of simple Lie algebras are Darboux integrable, that is, they admit complete families of essentially independent integrals along both characteristics. We consider semi-discrete and purely discrete analogs of these systems and prove their Darboux integrability which appears to be a direct consequence of the nature of Toda lattice related to Darboux-Laplace transformations.

If there is enough time, we will also discuss the notion of characteristic algebra which is an algebraic structure that controls the existence of characteristic integrals for a hyperbolic equation and its growth properties describe the behaviour of the corresponding hyperbolic system.