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.