Abstract. 
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
$\liminf_{n\to\infty}\limits n|T^nx-y|=0$.

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.

Several open problem (related) will be presented.