One of the techniques used in quantum field theory in curved space-times to eliminate divergences in the vacuum expectation value of the stress tensor for quantum fields propagating on a classical gravitational background is called covariant geodesic point separation. Beginning with the Schwinger-DeWitt proper-time method we show how to discard divergences in the effective action by renormalization of the coupling constants in a classical gravitational action functional. We then demonstrate how to determine which terms in the vacuum expectation value of the stress tensor vanish when this renormalization is carried out. This is done using the point-separation approach. We give the form of these terms for spin 0, 1/2, and 1 fields, massive or massless, on an arbitrary curved background. The procedure used is covariant and introduces no ambiguities beyond those inherent in any renormalization scheme. We note the appearance of trace anomalies which arise due to the breaking of conformal invariance by the renormalization process and give the form of the anomalies for arbitrary space-time dimension.