  Within a framework of noncommutative geometry, we develop an analogue of
(pseudo) Riemannian geometry on finite and discrete sets. On a finite set,
there is a counterpart of the continuum metric tensor with a simple geometric
interpretation. The latter is based on a correspondence between first order
differential calculi and digraphs. Arrows originating from a vertex span its
(co)tangent space. If the metric is to measure length and angles at some point,
it has to be taken as an element of the left-linear tensor product of the space
of 1-forms with itself, and not as an element of the (non-local) tensor product
over the algebra of functions. It turns out that linear connections can always
be extended to this left tensor product, so that metric compatibility can be
defined in the same way as in continuum Riemannian geometry. In particular, in
the case of the universal differential calculus on a finite set, the Euclidean
geometry of polyhedra is recovered from conditions of metric compatibility and
vanishing torsion. In our rather general framework (which also comprises
structures which are far away from continuum differential geometry), there is
in general nothing like a Ricci tensor or a curvature scalar. Because of the
non-locality of tensor products (over the algebra of functions) of forms,
corresponding components (with respect to some module basis) turn out to be
rather non-local objects. But one can make use of the parallel transport
associated with a connection to `localize' such objects and in certain cases
there is a distinguished way to achieve this. This leads to covariant
components of the curvature tensor which then allow a contraction to a Ricci
tensor. In the case of a differential calculus associated with a hypercubic
lattice we propose a new discrete analogue of the (vacuum) Einstein equations.
