Abstract: There is a long history in number theory of finiteness theorems for Diophantine equations, assertions that a certain equation has only finitely many solutions. Even further, one can ask for effective finiteness theorems; a theorem is effective when it gives an upper bound for the size of any solution, reducing the problem of listing all solutions to a finite (in principle!) computation. Effective finiteness theorems, unfortunately, are very hard to come by. But there is an interesting interrmediate goal: theorems which give upper bounds for the number of solutions, or for the number of solutions subject to some conditions. I will give a general talk about the recent history of results of this kind, avoiding the technical guts of things and trying to give some general idea of strategies, finishing with a recent result of mine with Lawrence and Venkatesh, and a theorem of (2017 Yale Ph.D.) Vesselin Dimitrov with Gao and Habegger.