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ABSTRACf This study is concerned with detailed expansions for the renewal function H(t) and various of its generalizations. Our assumptions concerning the growth rate of the tail of the underlying lifetime distribution F(x) are of the form f: x\lM(x)dF(x) < co, where \I ~ 0 and M(x) belongs to an appropriate class of monotone functions. l~en F(x) has a finite variance, it is shown that (for t > 0) t ~2 H(t) =-1 + :-z F(2) (t) + L(t) , ~l 2~1 where ~i denotes the ith moment of F(x), F(2) (x) is the second derived distribution of F(x) , and L(t) is a function of bounded variation such that, in particular, L(t) = o(l/tM(t)) as t ~ co. A similar expression for li(t) involving a finite number of derived distributions is derived for the infinite variance case. In addition our approach yields refined expansions for the factorial moments and cumulants of the number of renewals in the time interval (O,t). By developing estimates for the moments of the forward recurrence-time we can also evaluate the variance of the number of renewals in an interval of time away from the origin. Our main task throughout is to demonstrate that various remainder terms are functions B(x) of bounded variation belonging to some moment class B(M; \I) defined by tIle property f~colxl\l M(lxl)ldB(x) I < co. For this purpose we prove an extension of a theorem due to Wiener, Pitt, , Levy, and Smith concerning analytic filllctions of Fourier-Stieltjes transfonns of flUlctions in SCM; \I); our version is the most general which can be obtained with respect to M(x). We also demonstrate that certain convolutions which might be presumed to belong to SCM; 0) are , actually in SCM; 1). In conjlUlction with the Wiener-Pitt-Levy-Smith result this lUlexpected property (referred to as "smoothing magic") yields renewal theoretic results which are stronger than can be achieved using the fonner alone. Several of these theorems are applied in a discussion of the time-dependent behavior of the superposition of identical independent renewal processes. Aspects considered include the distribution of the number of events in a time interval near the origin, as well as the variance-time and covariance-time flUlctions.