In this talk I will briefly review Moser-Trudinger inequalities, and David Adams' important contributions to the subject. I will then talk about recent work with Luigi Fontana, where we extend, unify and improve Adams' results in an abstract, measure-theoretic setting. I will then present a few of the several applications of such results, including: sharp inequalities for general elliptic differential operators, sharp inequalities on the Heisenberg group and the CR sphere, the Beckner-Onofri inequality (in joint work with Branson and Fontana) and sharp inequalities without boundary conditions on the unit ball of R^n.