  Differential calculus on discrete sets is developed in the spirit of
noncommutative geometry. Any differential algebra on a discrete set can be
regarded as a `reduction' of the `universal differential algebra' and this
allows a systematic exploration of differential algebras on a given set.
Associated with a differential algebra is a (di)graph where two vertices are
connected by at most two (antiparallel) arrows. The interpretation of such a
graph as a `Hasse diagram' determining a (locally finite) topology then
establishes contact with recent work by other authors in which discretizations
of topological spaces and corresponding field theories were considered which
retain their global topological structure. It is shown that field theories, and
in particular gauge theories, can be formulated on a discrete set in close
analogy with the continuum case. The framework presented generalizes ordinary
lattice theory which is recovered from an oriented (hypercubic) lattice graph.
It also includes, e.g., the two-point space used by Connes and Lott (and
others) in models of elementary particle physics. The formalism suggests that
the latter be regarded as an approximation of a manifold and thus opens a way
to relate models with an `internal' discrete space ({\`a} la Connes et al.) to
models of dimensionally reduced gauge fields. Furthermore, also a `symmetric
lattice' is studied which (in a certain continuum limit) turns out to be
related to a `noncommutative differential calculus' on manifolds.
