Non-Kahler Calabi-Yau Manifolds

String theory has had a profound influence on research in Calabi-Yau spaces over the past twenty-five years. We first briefly mention some of the work in Kahler Calabi-Yau manifolds that was influenced by the discovery of mirror symmetry in the late 1980s. We then discuss some of the mathematical motivations behind the recent work on non-Kahler Calabi-Yau manifolds, which arise in string compactifications with fluxes. After extending mirror symmetry to non-Kahler Calabi-Yau manifolds, we show how this leads to new cohomologies and invariants of non-Kahler symplectic manifolds.