Let $f$ be a periodic measurable function and $(n_k)$ an increasing sequence of positive integers. The authors study conditions under which the series $\sum_{k=1}^\infty c_k f(n_kx)$ converges in mean and for almost every $x$. There is a wide classical literature on this problem going back to the 30's, but the results for general $f$ are much less complete than in the trigonometric case $f(x)=\sin x$. As it turns out, the convergence properties of $\sum_{k=1}^\infty c_k f(n_kx)$ in the general case are determined by a delicate interplay between the coefficient sequence $(c_k)$, the analytic properties of $f$ and the growth speed and number-theoretic properties of $(n_k)$. In this paper the authors give a general study of this convergence problem, prove several new results and improve a number of old results in the field. They also study the case when the $n_k$ are random and investigate the discrepancy the sequence $\{n_kx\}$ mod 1.