We will discuss the non-Kahler Calabi-Yau geometry introduced by string theorists C. Hull and A. Strominger. We propose to study these spaces via a parabolic PDE which is a nonlinear flow of non-Kahler metrics. This talk will survey works with T. Collins, T. Fei, D.H. Phong, S.-T. Yau, and X.-W. Zhang.