In this thesis, a novel and efficient numerical method is presented for the computation of the L² optimal mass transport mapping in two and three dimensions. The method uses a direct variational approach. A new projection to the constraint technique has been formulated that can yield a good starting point for the method as well as a second order accurate discretization to the problem. The numerical experiments demonstrate that the algorithm yields accurate results in a relatively small number of iterations that are mesh independent. In the first part of the thesis, the theory and implementation details of the proposed method are presented. These include the reformulation of the Monge-Kantorovich problem using a variational approach and then using a consistent discretization in conjunction with the "discretize-then-optimize" approach to solve the resulting discrete system of differential equations. Advanced numerical methods such as multigrid and adaptive mesh refinement have been employed to solve the linear systems in practical time for even 3D applications. In the second part, the methods efficacy is shown via application to various image processing tasks. These include image registration and morphing. Application of (OMT) to registration is presented in the context of medical imaging and in particular image guided therapy where registration is used to align multiple data sets with each other and with the patient. It is shown that an elastic warping methodology based on the notion of mass transport is quite natural for several medical imaging applications where density can be a key measure of similarity between different data sets e.g. proton density based imagery provided by MR. An application is also presented of the two dimensional optimal mass transport algorithm to compute diffeomorphic correspondence maps between curves for geometric interpolation in an active contour based visual tracking application.