Main subject categories: • Stationary PDEs • Differential equations • PDEs • Semigroups and Evolution Equations: Functional Calculus, Regularity and Kernel Estimates • Front Tracking Method for Systems of Conservation Laws • Current Issues on Singular and Degenerate Evolution Equations • Nonlinear Hyperbolic–Parabolic Coupled Systems • Nonlinear Parabolic Equations and Systems • Kinetic Formulations of Parabolic and Hyperbolic PDEs: From Theory to Numerics • L¹-stability of Nonlinear Waves in Scalar Conservation LawsThe aim of this Handbook is to acquaint the reader with the current status of the theory of evolutionary PDEs, and with some of its applications. This is not an easy task: Unlike other mathematical theories which exhibit a tree-like structure, with clearly distinguishable trunk, main and secondary branches, the theory of PDEs has the appearance of a bush with very complex structure. Its roots as well as its flowers are often related to the physical world, and it is fertilized by ideas and techniques borrowed from virtually every other area of mathematics.Evolutionary PDEs made their first appearance in the 18th century, in the endeavor to understand the motion of fluids and other continuous media. It is remarkable that this research program is still ongoing and many fundamental questions remain unanswered. Beyond this area, however, evolutionary PDEs have become ubiquitous, as they seem to govern the dynamics of any physical, chemical, biological, ecological or economic system whose state is described by spatially dependent variables.Volume I of this Handbook will focus on the abstract theory of evolutionary equations, addressing questions of existence, uniqueness and other general qualitative properties of solutions. Future volumes will consider more concrete problems relating to specific applications. Our hope is that the handbook will provide a panorama of this amazingly complex and rapidly developing branch of mathematics.