Let M be a complete non-compact Riemannian manifold, or more generally a metric measure space endowed with a heat kernel, satisfying the volume doubling property. In this survey, the connection between various kinds of upper and lower heat kernel estimates will be examined. We will explain the connections between heat kernel estimates and Sobolev inequalities, some sufficient conditions in terms of heat kernel gradient estimates for $L^p$-boundedness of the Riesz transform, and finally, in the polynomial growth setting, the connection between boundedness of Riesz transform and a new version of Sobolev type inequalities.