  We present a formulation of Regge Calculus where arbitrary coordinates are
associated to each vertex of a simplicial complex and the degrees of freedom
are given by the metric on each simplex. The lengths of the edges are thus
determined and are left invariant under arbitrary transformations of the
discrete set of coordinates, provided the metric transforms accordingly.
Invariance under coordinate transformations entails tensor calculus and our
formulation follows closely the usual formalism of the continuum theory. The
definitions of parallel transport, Christoffel symbol, covariant derivatives
and Riemann curvature tensor follow in a rather natural way. In this
correspondence Einstein action becomes Regge action with the deficit angle
$\theta$ replaced by $\sin \theta$. The correspondence with the continuum
theory can be extended to actions with higher powers of the curvature tensor,
to the vielbein formalism and to the coupling of gravity with matter fields
(scalars, fermionic fields including spin $3/2$ fields and gauge fields) which
are then determined unambiguously and discussed in the paper. An action on the
simplicial lattice for $N=1$ supergravity in $4$ dimensions is derived in this
context. Another relavant result is that Yang-mills actions on a simplicial
lattice consist, even in absence of gravity, of two plaquettes terms, unlike
the one plaquette Wilson action on the hypercubic lattice. An attempt is also
made to formulate a discrete differential calculus to include differential
forms of higher order and the gauging of free differential algebras in this
scheme. However this leads to form products that do not satisfy associativity
and distributive law with respect to the $d$ operator. A proper formulation of
theories that contain higher order differential forms in the context of Regge
Calculus is then still lacking.
