For an arbitrary finite group G and for any generator set X of G,one can define the Cayley graph Cay(G,X). 

A widely studied topic is the examination of the growth of such graphs. Such a question is a conjecture of Babai, stating that if G is a finite simple group then the diameter of every Cayley graph of G is bounded by a polynomial function of log|G|. 

Further problems of this kind are to find conditions, which helpto define new expander families of Cayley graphs, and to searchfor versions of the product theorem of Pyber-Szabó and Breuillard-Green-Tao for groups of Lie type of unbounded rank.