  `Umbral calculus' deals with representations of the canonical commutation
relations. We present a short exposition of it and discuss how this calculus
can be used to discretize continuum models and to construct representations of
Lie algebras on a lattice. Related ideas appeared in recent publications and we
show that the examples treated there are special cases of umbral calculus. This
observation then suggests various generalizations of these examples. A special
umbral representation of the canonical commutation relations given in terms of
the position and momentum operator on a lattice is investigated in detail.
