Classical work of Zygmund, Salem, Erdoes and Gal is devoted to the probabilistic analysis of trgonometric sums satisfying the so-called Hadamard gap condition. Later (1975) this investigation was extended to discrepancies of exponentially growing sequences culminating in W. Philipp`s proof of a law of the iterated logarithm. In the present talk the developments in this area during the last 50 years is outlined: central limit theorems and precise asymptotic expansions are established. Furthermore, some sublacunary sequences are discussed, e.g. the so-called Hardy-Littlewood-Polya sequence. This involves methods from Diophantine analysis such as the subspace theorem and generalizations. The lecture is based on recent joint work of C. Aistleitner, I. Berkes and R. F. Tichy, published in the SMF series "Panoramas and Synthèses", vol. 62.