This is a talk at the New York Graduate Student Logic Conference on 13 May 2016.
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.