A survey on the (2,3)-generation problem and related topics

A $(2,3)$-generated group is a group that can be generated by an involution and an element of order 3. These groups together with the trivial one and two cyclic groups of orders 2 and 3 are exactly the quotients of the modular group ${\rm PSL}(2,{\mathbb Z})$.
During past decades there was a considerable progress in determining which finite simple groups are (2,3)-generated. Using probabilistic methods Liebeck and Shalev showed that almost all finite classical groups of large rank are (2,3)-generated. However, the full list of exceptions is still unknown. Another (constructive) approach was developed by many authors. Recently Pellegrini filled the last gaps for the series ${\rm PLS(n,q)}$ and Tamburini and Pellegrini completed the unitary and symplectic cases.
Hurwitz groups (or finite (2,3,7)-generated groups) form an important subclass of the (2,3)-generated groups. In general, the situation is quite similar, i.e., most of the finite simple groups of large rank are Hurwitz, but for small ranks we know less.
In my talk I will survey main results, open questions and methods used in this area.