The (real) Monge-Ampère equation is a fully nonlinear elliptic equation with a strong geometric nature. It can indeed be used, following Minkowski, to recover a convex hypersurface from the knowledge of its Gaussian curvature. This equation also plays a central role in the theory of optimal transport, which has a wide range of applications in physics, economics and geometry. However, the singular and highly non-linear nature of the equation is a serious obstruction to the development of efficient solvers. We propose to attack the problem from an original side, that combines the point of views of analysts and computational geometers, both categories being represented by members of this project. First, we will develop and study solvers for the standard Monge-Ampère equation in dimension two and three. Our objective is that it becomes possible to deal with very large discretizations without parameter tweaking. This will allow their use for the numerical resolution of difficult non-linear minimization problems involving Monge-Ampère operators. This includes minimization problems over the space of convex bodies or of convex functions but also minimization problems formulated in term of optimal transport. For instance, we will consider the problem of approximating minimizing geodesics over the space of measure-preserving maps, which can be used to construct non-deterministic solutions to Euler's equation for incompressible fluids. In all these cases, we hope that numerical experiments will lead to a better mathematical understanding of the underlying problems. Second, we will extend the computational geometric approach to generalized Monge-Ampère equations arising in particular from non-imaging optics. This includes the problem of designing reflectors and refractors that transport the light energy emitted from a point source into a target whose shape and intensity distribution is prescribed. Some of the equations occuring from non-imaging optics belong to a more general class of equations called ``prescribed generated Jacobian equations'', whose discretization will require new theoretical tools and sophisticated geometric algorithms.