Mutual Information belongs to the most popular and powerful tools for multi-modal image registration. Nevertheless, its handling is involved, and despite a wide spread use in a large number of applications no standard treatment exists. Here, we present both theory and numerics for non-linear image registration based on mutual information. Building on a non-standard analysis of mutual information from the perspective of measure and integration theory, we develop PDE and optimization based registration methods. Crucial for the numerical schemes are efficient computation of mutual information and fast solving of the arising large linear systems. Addressing these points, we consider non-equispaced Fourier transforms and B-Splines techniques for the approximation of densities and develop fast FFT and multigrid solvers. Finally, we demonstrate the performance of the proposed methods in an application for the registration of 3D PET and CT images.