Convex geometry is the study of convex bodies in Euclidean space. Despite the apparent simplicity of such objects, they are a source of many deep mathematical discoveries and mysteries. This talk will present a survey of Brunn-Minkowski theory, which is the study of affine geometric invariants and inequalities satisfied by convex bodies. Unlike differential geometry, the invairants are usually integral invariants. The inequalities are generalized isoperimetric inequalities and, surprisingly, reverse isoperimetric inequalities. Other fundamental questions involve generalizations of the Minkowski problem and, if sufficient regularity is assumed, equivalent to Monge-Ampere type PDE’s.