  We discuss umbral calculus as a method of systematically discretizing linear
differential equations while preserving their point symmetries as well as
generalized symmetries. The method is then applied to the Schr\"{o}dinger
equation in order to obtain a realization of nonrelativistic quantum mechanics
in discrete space-time. In this approach a quantum system on a lattice has a
symmetry algebra isomorphic to that of the continuous case. Moreover, systems
that are integrable, superintegrable or exactly solvable preserve these
properties in the discrete case.
