This book introduces the treatment of linear and nonlinear (quasi-linear) abstract evolution equations by methods from the theory of strongly continuous semigroups. The theoretical part is accessible to graduate students with basic knowledge in functional analysis, with only some examples requiring more specialized knowledge from the spectral theory of linear, self-adjoint operators in Hilbert spaces. Emphasis is placed on equations of the hyperbolic type which are less often treated in the literature. Semigroup Theory uses abstract methods of Operator Theory to treat initial bou- ary value problems for linear and nonlinear equations that describe the evolution of a system. Due to the generality of its methods, the class of systems that can be treated in this way exceeds by far those described by equations containing only local op- ators induced by partial derivatives, i.e., PDEs. In particular, that class includes the systems of Quantum Theory. Another important application of semigroup methods is in ?eld quantization. Simple examples are given by the cases of free ?elds in Minkowski spacetime like Klein-Gordon ?elds, the Dirac ?eld and the Maxwell ?eld, whose ?eld equations are given by systems of linear PDEs. The second quantization of such a ?eld replaces the ?eld equation by a Schrodinger ¿ equation whose Hamilton operator is given by the second quantization of a non-local function of a self-adjoint linear operator. That operator generates the semigroup given by the time-development of the solutions of the ?eld equation corresponding to arbitrary initial data as a function of time. Conventions 1(5) Mathematical Introduction 5(8) Quantum Theory 5(3) Wave Equations 8(5) Prerequisites 13(28) Linear Operators in Banach Spaces 13(12) Weak Integration of Banach Space-Valued Maps 25(10) Exponentials of Bounded Linear Operators 35(6) Strongly Continuous Semigroups 41(30) Elementary Properties 42(9) Characterizations 51(7) An Integral Representation in the Complex Case 58(1) Perturbation Theorems 59(4) Strongly Continuous Groups 63(3) Associated Inhomogeneous Initial Value Problems 66(5) Examples of Generators of Strongly Continuous Semigroups 71(34) The Ordinary Derivative on a Bounded Interval 71(3) Linear Stability of Ideal Rotating Couette Flows 74(3) Outgoing Boundary Conditions 77(7) Damped Wave Equations 84(13) Autonomous Linear Hermitian Hyperbolic Systems 97(8) Intertwining Relations, Operator Homomorphisms 105(18) Semigroups and Their Restrictions 105(9) Intertwining Relations 114(3) Nonexpansive Homomorphisms 117(6) Examples of Constrained Systems 123(14) 1-D Wave Equations with Sommerfeld Boundary Conditions 123(4) 1-D Wave Equations with Engquist-Majda Boundary Conditions 127(5) Maxwell's Equations in Flat Space 132(5) Kernels, Chains, and Evolution Operators 137(28) A Convolution Calculus with Operator-Valued Kernels 138(8) Chains 146(1) Juxtaposition of Chains 147(1) Finitely Generated Chains 148(1) Evolution Operators 149(6) Stable Families of Generators 155(10) The Linear Evolution Equation 165(12) Examples of Linear Evolution Equations 177(38) Scalar Fields in the Gravitational Field of a Spherical Black Hole 178(21) Non-Autonomous Linear Hermitian Hyperbolic Systems 199(16) The Quasi-Linear Evolution Equation 215(20) Examples of Quasi-Linear Evolution Equations 235(30) A Generalized Inviscid Burgers' Equation 235(11) Quasi-Linear Hermitian Hyperbolic Systems 246(19) Appendix 265(4) References 269(10) Index of Notation 279(2) Index of Terminology 281