Abstract : The theory of cumulative processes, introduced and developed by W. L. Smith, provides a significant generalization of the renewal counting process. One examines the question of extending the Blackwell and key renewal theorems to cumulative processes. For a subclass of cumulative processes, which one calls strongly cumulative, the Blackwell and key renewal theorems hold under very general conditions. This class of cumulative processes includes all the standard examples of cumulative processes. One also studies processes of the form Y(t) = the integral from o to t V(s)ds where V is a regenerative process. Smith has shown that under mild conditions V(s) converges in distribution, say to V(infinity), as s approaches infinity, and that Y(t)/t converges almost surely and in expectation to Kappa sub 1/mu sub 1 (a constant). The result is that Kappa sub 1/mu sub 1 = EV(infinity). This holds even if lim as t approaches infinity, of EV(t) does not exist.