In this talk I will review “mirror symmetry” and outline a generalization currently being developed. Mirror symmetry is a prediction of physics, in particular string theory. It is a duality between often-topologically-distinct complex Kähler manifolds, one which exchanges relatively hard curve-counting computations with relatively easy computations of periods in complex geometry. Understanding mirror symmetry allows one to make predictions for enumerative geometry, and led to a revolution in parts of algebraic geometry several years ago. Heterotic mirror symmetry is an attempted generalization which exchanges pairs (X, E), (X’, E’) where X and X’ are complex Kähler manifolds, and E → X, E’ → X’ are holomorphic vector bundles. We shall review ordinary mirror symmetry as an example of a very fruitful interdisciplinary mathematics/physics interaction, and also outline heterotic mirror symmetry.