In these lecture notes, I will describe heterotic/type-II duality in six and four dimensions. When supersymmetry is the maximalN=4it will be shown that the duality reduces in the field theory limit to the Montonen-Olive duality ofN=4Super Yang-Mills theory. We will consider further compactifications of type II theory on Calabi-Yau manifolds. We will understand the physical meaning of geometric conifold singularities and the dynamics of conifold transitions. When the CY manifold is a K3 fibration we will argue that the type-II ground-state is dual to the heterotic theory compactified onK3×T2.This allows an exact computation of the low effective action. Taking the field theory limit,&agr;′→0,we will recover the Seiberg-Witten non-perturbative solution ofN=2gauge theory. ©1999 American Institute of Physics.