  In this paper we define and study a notion of discrete homology theory for
metric spaces. Instead of working with simplicial homology, our chain complexes
are given by Lipschitz maps from an $n$-dimensional cube to a fixed metric
space. We prove that the resulting homology theory verifies a discrete analogue
of the Eilenberg-Steenrod axioms, and prove a discrete analogue of the
Mayer-Vietoris exact sequence. Moreover, this discrete homology theory is
related to the discrete homotopy theory of a metric space through a discrete
analogue of the Hurewicz theorem. We study the class of groups that can arise
as discrete homology groups and, in this setting, we prove that the fundamental
group of a smooth, connected, metrizable, compact manifold is isomorphic to the
discrete fundamental group of a `fine enough' rectangulation of the manifold.
Finally, we show that this discrete homology theory can be coarsened, leading
to a new non-trivial coarse invariant of a metric space.
