A Basic Analysis Toolbox Some Basic Mathematical Shorthand Set Algebra Functions The Space (R, ¿) Sequences in (R, ¿) The Spaces (R N , ¿R N ) and (M N (R), ¿M N (R)) Abstract Spaces Elementary Calculus in Abstract Spaces Some Elementary ODEs Looking Ahead Guidance for Exercises Homogenous Linear Evolution Equations in R N Motivation by Models The Matrix Exponential The Homogenous Cauchy Problem: Well-Posedness Perturbation and Convergence Results A Glimpse at Long-Term Behavior Looking Ahead Guidance for Exercises Abstract Homogenous Linear Evolution Equations Linear Operators Motivation by Models Introducing Semigroups The Abstract Homogenous Cauchy Problem Generation Theorems A Useful Perturbation Result Some Approximation Theory A Brief Glimpse at Long-Term Behavior An Important Look Back Looking Ahead Guidance for Exercises Nonhomogenous Linear Evolution Equations Finite-Dimensional Setting Infinite-Dimensional Setting Introducing Two New Models Looking Ahead Guidance for Exercises Semi-Linear Evolution Equations Motivation by Models More Tools from Functional Analysis Some Essential Preliminary Considerations Growth Conditions Theory for Lipschitz-Type Forcing Terms Theory for Non-Lipschitz-Type Forcing Terms Theory under Compactness Assumptions A Summarizing Look Back Looking Ahead Guidance for Exercises Functional Evolution Equations Motivation by Models Functionals Theory in the Lipschitz Case Theory under Compactness Assumptions Models¿New and Old Looking Ahead Guidance for Exercises Implicit Evolution Equations Sobolev-Type Equations Neutral Evolution Equations Looking Ahead Guidance for Exercises Delay Evolution Equations Motivation by Models Setting and Formulation of the Problem Theory for Lipschitz-Type Forcing Terms Theory for Non-Lipschitz-Type Forcing Terms Implicit Delay Evolution Equations Other Forms of Delay Models¿New and Old An Important Look Back! Looking Ahead Guidance for Exercises Nonlinear Evolution Equations A Wealth of New Models Comparison of the Linear and Nonlinear Settings The Crandall¿Liggett Theory A Quick Look at Nonlinear Evolution Inclusions Some Final Comments Guidance for Exercises Nonlocal Evolution Equations Introductory Remarks Motivation by Models Some Abstract Theory Final Comments Beyond Volume 1¿ Three New Classes of Evolution Equations Next Stop¿ Stochastic Evolution Equations!: The Preface to Volume 2 Bibliography Index.