Character methods, probabilistic methods, and their combination, have been instrumental in the solution of various problems in group theory over the years.

In the talk I will provide a brief background and then focus on some brand new results and conjectures, related to random walks and to word maps.

In particular I will discuss a solution (joint work with Larsen and Tiep) of the so-called Probabilistic Waring Problem for finite simple groups, which applies strong new character bounds.