This is a talk at the Boise Extravaganza in Set Theory, part of the 97th annual meeting of the AAAS Pacific Division on 15 June 2016. (BEST originated, as one might expect, in Boise, Idaho, but later went on the road.)
A well-known result is that there is a minimal transitive model of ZFC. In this talk I look at the analogous question for second-order set theories. That there is a minimal transitive model of GBC follows immediately from the result from ZFC but the KM case is more difficult. The main result I will present is that the question for KM has a negative answer: there is no least transitive model of KM. Along the way, we will look at another notion of minimality for models of second-order set theory and see that KM does not have minimal models in this other respect.