We explain recent results on stability theorems for some classical functional and ge- ometric inequalities, along with two applications: one to evolution equations, and one to statistical mechanics. The inequalities in question include certain Gagliardo-Nirenberg- Sobolev inequalities, the Brun-Minkowski inequality, for example. In these inequalities, all of the cases of equality are known, and indeed, Minkowski's contribution to the Brun- Minkowski inequality was to both determine the cases of equality. One can now ask if, in such an inequality, one almost has equality, is one in some sense near to one of the known cases of equality? A stability theorem is a theorem that provides a positive answer to this sort of question, and as indicated above, we shall explain and sketch the proofs of several such results, and we shall also explain two of the applications that motivated these investigations, which were carried out in collaboration with Alessio Figalli and Francesco Maggi. Though we shall keep the discussion of the applications non-technical as bets a colloquium talk, we nonetheless hope to convey an understanding of why it might be very useful to solve some of the many open problems in this field.