The optimal transport problem was proposed by Gaspard Monge in 1781 to find the cheapest way of transporting materials from one location to another. Owing to its highly nonlinear nature, this problem remained impenetrable for over 150 years, and a satisfactory mathematical theory has emerged only recently due to the combined efforts of various researchers. Finding efficient ways of computing optimal transport maps is also a vibrant topic of research in applied analysis, spurred by myriad applications to the physical and social sciences. In this talk, I will survey a bit of the history and mathematical aspects of the optimal transport problem and discuss a dynamic approach for finding optimal maps using a time-dependent Monge-Ampere equation. The results I will present were obtained in collaboration with Jun Kitagawa (Michigan State University).