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.