This comprehensive two-volume textbook presents the whole area of Partial Differential Equations - of the elliptic, parabolic, and hyperbolic type - in two and several variables. Special emphasis is put on the connection of PDEs and complex variable methods. In this second volume the following topics are treated: Solvability of operator equations in Banach spaces, Linear operators in Hilbert spaces and spectral theory, Schauder's theory of linear elliptic differential equations, Weak solutions of differential equations, Nonlinear partial differential equations and characteristics, Nonlinear elliptic systems with differential-geometric applications. While partial differential equations are solved via integral representations in the preceding volume, functional analytic methods are used in this volume. This textbook can be chosen for a course over several semesters on a medium level. Advanced readers may study each chapter independently from the others.