This dissertation deals with different aspects of numerical and mathematical analysis of systems of possibly degenerate partial differential equations. Under particular conditions, solutions to these equations in the considered applications exhibit steep gradients, and in the degenerate case, sharp fronts and discontinuities. This calls for a concentration of computational effort in zones of strong variation. To achieve this goal we introduce suitable finite volume methods and fully adaptive multiresolution schemes for spatially one, two and three-dimensional, possibly degenerate reaction-diffusion systems, focusing on sedimentation processes in the mineral industry and traffic flow problems, two and three-dimensional reaction-diffusion systems modeling population dynamics, combustion processes, cardiac propagation and models of pattern formation and chemotaxis in mathematical biology. We are also interested in the study of convergence of the finite volume approximations, and the wellposedness and regularity analysis of weak solutions.