The existence, uniqueness and regularity of the solutions of systems of partial differential equations are analyzed. The formation of singularities and patterns, the evolution of certain geometric objects and the concentration of energy in localized structures are also studied. The modelling of physical phenomena by means of systems of equations is also one of the objectives, as well as simplifying existing models with the aid of rigorous approximations and asymptotic analyses. The focus is on the analysis of non-linear equations.