We consider the random group given by the quotient of the free group on n generators by n random relations. We discuss the proof of convergence of these groups to a limiting group as n goes to infinity, which includes two major parts. The existence of limiting behavior comes about from counting theorems in group theory. To prove there is no escape of mass in the limit involves analysis on the space of groups. We explain analogs of this construction that give random groups that we conjecture to model the Galois groups of the maximal unramified extensions of random number fields. This work is joint with Yuan Liu. The bulk of the talk will require basic background in group theory and probability (quotients of free groups, random variables); the part on the connection to number theory will require basic algebraic number theory (unramified extensions, inertia groups).