Measure transportation has rich and profound content in pure, applied and computational mathematics, with successful application in many field, ranging from fluid dynamics to engineering, economics, operations research, probability, and machine learning. The main objective of this PhD project is to develop a deep mathematical understanding of the measure transportation and to apply to a number of practical problems in machine learning. There are three particular objectives as follows:Transform copulas by mass transportation; Redistribution as divergence-free movement; Derive measures similar to the Wasserstein distances.