A suitably coherent family of finite groups can be ‘stuck together’ to make a compact (infinite) group, by forming an ‘inverse limit’. The resulting object is called a profinite group. For example, the family (Z/p^nZ) (n>0) corresponds to the ring of p-adic integers Z_p. All sorts of interesting questions about infinite groups, or about infinite families of finite groups, can be approached by studying the associated profinite group. I will give various illustrations.