This book is devoted to a large class of partial differential equations and systems which are nonlinear in the highest derivatives. The authors present a new functional analytic method based on the Baire category theorem for handling the existence of almost everywhere solutions of these equations. Comparison with other methods is discussed: essentially that of viscosity solutions, but also briefly that of convex integration. Results obtained by this new method have important applications to the calculus of variations, geometry, nonlinear elasticity, problems of phase transitions and optimal design. The book is divided into four parts. Part I examines first and second order partial differential equations while Part II considers systems. Building on the theory presented, Part III is devoted to applications, including the singular values case, the case of potential wells, and the complex eikonal equation. In Part IV the authors gather some nonclassical Vitali type covering theorems, as well as several fine results on the approximation of Sobolev functions by piecewise affine or polynomial functions. These results have relevance in other contexts, such as numerical analysis. This monograph is intended for advanced graduate students and researchers in nonlinear analysis and its applications. The book is essentially self-contained and contains many mathematical examples derived from applications to the materials sciences.