I will begin by explaining the fundamental construction of a convex toric degeneration which has been rediscovered by many people and which appears in different guises in many fields of mathematics. I will illustrate it by giving examples of toric degenerations as varieties, as pairs, as cycles, and as subschemes. I will also explain the extension of this construction to degenerations of abelian varieties, and of hyperplane arrangements. In the second talk, I will apply the fundamental construction to making various combinatorial moduli spaces: of stable toric pairs, toric Hilbert scheme, toric Chow varieties, stable semiabelic varieties, and stable hyperplane arrangements. Time permitting, I will go over applications of the theory of degenerations of abelian varieties to a problem of extending various Torelli maps to compactified moduli spaces: for Jacobians, Prym varieties, Prym-Tyurin varieties, and Intermediate jacobians.