TASI Lectures on Compatification and Duality

We describe the moduli spaces of theories with 32 or 16 supercharges, from several points of view. Included is a review of backgrounds with D-branes (including type I' vacua and F-theory), a discussion of holonomy of Riemannian metrics, and an introduction to the relevant portions of algebraic geometry. The case of K3 surfaces is treated in some detail.