Infinite index subgroups of integer matrix groups like SL(n,Z) which are Zariski dense in SL(n), arise in geometric diophantine problems (eg Integral Apollonian packings), as monodromy groups associated with families of algebraic varieties, as reflection groups... One of the key features needed when applying such groups in number theoretic problems is that the congruence graphs associated with these groups are "expanders". We will introduce and explain these ideas and review some recent developments and applications.