One of the fundamental results of probability is the Hartman-Wintner (1941) law of the iterated logarithm (LIL) which holds for random variables with finite variance and mean zero. It is natural to ask whether one can obtain related results in the infinite variance case. The corresponding problem in connection with the central limit theorem has been solved in the early 50's, but though there has been some progress for the LIL problem in the last years, there are still many open questions, even for distributions which appear quite simple. In the first part of this talk we shall look at a slightly modified form of this general LIL problem and we shall show that one can generalize the classical LIL to a ``law of a very slowly varying function''. Using this result we a can solve the original problem also for some distributions which have proven quite resilient in the past. In the second part of the talk we shall give a survey of some related results. In particular, we shall present a functional version of our law of a very slowly varying function.