Preliminary general material -- Steady-State Problems -- Pseudomonotone or weakly continuous mappings -- Accretive mappings -- Potential problems: smooth case -- Nonsmooth problems; variational inequalities -- Systems of equations: particular examples -- Evolution Problems -- Special auxiliary tools -- Evolution by pseudomonotone or weakly continuous mappings -- Evolution governed by accretive mappings -- Evolution governed by certain set-valued mappings -- Doubly-nonlinear problems -- Systems of equations: particular examples.This book primarily concerns quasilinear and semilinear elliptic and parabolic partial differential equations, inequalities, and systems. It balances the abstract functional-analysis approach based on nonlinear monotone, pseudomonotone, weakly continuous, or accretive mappings with concrete partial differential equations in their weak (or more general) formulation. Methods of Galerkin or of Rothe are exposed in a large generality. Other methods include various direct methods, regularization, or fixed points. The exposition leads general theory as fast as possible towards the analysis of concrete equations, which have specific applications in continuum (thermo-) mechanics of solids and fluids, electrically (semi-) conductive media, modelling of biological systems, or in mechanical engineering. Selected parts are rather an introduction into the subject while some others form an advanced textbook. The intended audience is graduate and PhD students and researchers in the theory of partial differential equations or in mathematical modelling of distributed parameter systems.