This book contains several introductory texts concerning the main directions in the theoryof evolutionary partial differential equations. The main objective is to present clear, rigorous,and in depth surveys on the most important aspects of the present theory. The table ofcontents includes:W.Arendt: Semigroups and evolution equations: Calculus, regularity and kernel estimatesA.Bressan: The front tracking method for systems of conservation lawsE.DiBenedetto, J.M.Urbano,V.Vespri: Current issues on singular and degenerate evolution equations;L.Hsiao, S.Jiang: Nonlinear hyperbolic-parabolic coupled systemsA.Lunardi: Nonlinear parabolic equations and systemsD.Serre:L1-stability of nonlinear waves in scalar conservation lawsB.Perthame:Kinetic formulations of parabolic and hyperbolic PDE's: from theory to numerics"