Abstract: Let G be a finitely generated group and let G>G_1>G_2 ... be a sequence of finite index normal subgroups of G with trivial intersection.
We expect that the asymptotic behaviour of various group theoretic invariants of the groups G_i should relate to algebraic, topological or measure theoretic properties of G.
A classic example of this is the Luck approximation theorem which says that the growth of the ordinary Betti numbers of sequence G_i is given by the L^2-Betti number of (the classifying space) of G.
Another example is a theorem of Abert and myself that the growth of the ranks (=minimal number of generators) of $G_i$ is given by the cost of the action of G in the profinite completion of G with respect to the chain (G_i). Work of Lackenby relates growth of ranks of the sequence (G_i) with virtual splittings of G as an amalgam or an HNN extension. Other very interesting invariants to study are the growth of deficiency, and growth of torsion of the integral homology of the groups $G_i$. In this survey talk I will focus on the highlights of whats is known so far and on the many exciting open questions in this subject.