n this talk, we introduce and study some basic questions regarding the phase diagram for general mixed p-spin glasses. We focus on the analysis of the Parisi functional, a non-local, convex functional whose value is given by the solution of a certain Hamilton-Jacobi-Bellman PDE. We present an exact characterization of minimizers, and use this to study the phase diagram in the temperature-external field plane. In particular, we present some recent progress on the Replica Symmetric-to-Replica Symmetry Breaking phase transition: we prove that a natural generalization of the de Almeida-Thouless line does give the RS-RSB phase transition in the temperature-external field plane less a compact set. Time permitting, we will discuss some related work on the complexity of the Replica Symmetry Breaking phase for spherical mixed p-spin glasses. These are joint works with Aukosh Jagannath (Univ. of Toronto). Speaker(s): Ian Tobasco (UM)