This lecture is intended to be a survey of results of the type in the title. Historically, these ideas seem to have come to a head with the work of N. Varopoulous in the mid 1980’s culminating with the very nice little book: Geometry and Analysis on Groups, by Varopoulous, Saloff-Coste, Coulhon Cambridge U. Press, 1993. Other more recent work includes that of Saloff-Coste, Hebey, and others. Also, I will try to do this without dwelling on the definition/properties of Riemannian manifolds, but mention some of the major theorems about such that we need to formulate the results. Also, I’ll try to discuss some nonlinear PDE on smooth R-manifolds as time permits.