The classical shrinking target problem concerns the following set-up: Given a dynamical system (T, X) and a sequence of targets (B_n) of X, we investigate the size of the set of points x of X for which T^n(x) hits the target B_n for infinitely many n. There are many natural variants of this problem, some of which are much harder than the original shrinking target problem and require a vastly different toolkit. For example, one might be interested in the size of the liminf set of eventually always hitting points; or the size of a dynamical covering set, where instead of pre-images of balls we consider a limsup set of balls around orbit points.

In this talk I will discuss some of these problems in the context of iterated function systems, also covering some of my own past shrinking target results, joint with subsets of {Simon Baker, Thomas Jordan, Lingmin Liao, Michal Rams}. Time permitting, and if the results are ready on time, hot-off-the-press results on dynamical covering sets might also be explored (joint with Balazs Barany and Sascha Troscheit).