Word maps and finite simple groups

Any word in the free group on $d$ generators induces a word map from $G^d$ to $G$, for a given group $G$.

For finite simple groups, the following questions naturally arise:
1) Is this map surjective?
2) How the fibers of this map look like?
These questions have been investigated by Larsen, Liebeck, O'Brien,
Shalev, Tiep, the speaker and others.

For example, the Ore Conjecture, which had been open for more than 50 years, states that the commutator map is surjective on all finite simple groups.

In the talk, I will overview the known results in this area regarding  finite simple groups, and in particular, some recent results regarding  the group PSL(2,q) which were obtained in a joint work with Tatiana  Bandman and Fritz Grunewald.