  Simplicial approximation and the ideas associated with the Regge
calculus.provide a concrete way of implementing a sum over histories
formulation ofquantum gravity. A four-dimensional simplicial geometry is made
up of flat four-simplices joined together.A sum over simplicial geometries is a
sum over thedifferent ways the simplices can be joined together with an
integral over their edge lengths.Theconstruction of the simplicial Euclidean
action for this approach to quantum general relativity is illustrated. The
recovery of the diffeomorphism group in the continuum limit is discussed. Some
possible classes of simplicial complexes with which to define a sum over
topologies are described.
