The lecture is based on a recent preprint with the same title, joint with J. Chaika and put recently on arXiv. One of the results is that for ergodic IETs (Interval Exchange Transformations) almost sure $\liminf_{n\to\infty} n|T^nx-y|=0$. The result is optimal in two ways: \\ (1) the normalizing factor \ $n$ \ cannot be improved, even for rotations;\\ (2) the assumption of ergodicity cannot be replaced by just minimality.