  It is well known that every finite simple group can be generated by two
elements and this leads to a wide range of problems that have been the focus of
intensive research in recent years. In this survey article we discuss some of
the extraordinary generation properties of simple groups, focussing on topics
such as random generation, $(a,b)$-generation and spread, as well as
highlighting the application of probabilistic methods in the proofs of many of
the main results. We also present some recent work on the minimal generation of
maximal and second maximal subgroups of simple groups, which has applications
to the study of subgroup growth and the generation of primitive permutation
groups.
