Optimal mass transportation theory was originally developed as a way of solving optimization problems in the areas of operations research, probability and statistics. However, as recent applications of this theory have shown, certain analytical and geometric problems on the Euclidean space can effectively be treated in the corresponding Wasserstein space of probability measures. This project aims to study a class of dynamical systems on the Wasserstein space of probability measures corresponding to some fundamental systems of partial differential equations in fluid and quantum mechanics. A nice feature of the Wasserstein space approach is its ability to handle singular data and singular solutions. In addition, there is also a possibility of handling discrete and continuous models with the same formalism. The core of this program lies in the development of an appropriate theory for calculus of variations and Hamilton-Jacobi equations in the Wasserstein space.