This book provides an elementary introduction to the theory of observation and control for operator semigroups, focusing on admissibility, observability, and controllability. It is intended for readers with no prior knowledge of finite-dimensional control theory or operator semigroups. The book assumes a basic understanding of bounded operators on Hilbert spaces, differential equations, Fourier and Laplace transforms, distributions, and Sobolev spaces. Much of the necessary background is provided in the appendices. The book discusses the evolution of systems modeled by linear partial differential equations (PDEs) or linear delay differential equations, which can be described by operator semigroups. The state of such systems is an element in an infinite-dimensional normed space. The main topics include admissibility, observability, controllability, stabilizability, and detectability. The book covers various concepts of controllability and observability, with a focus on exact controllability and exact observability. Exact controllability is important because it guarantees stabilizability and the existence of a linear quadratic optimal control. Exact observability guarantees the existence of an exponentially converging state estimator and the existence of a linear quadratic optimal filter. The book also discusses other concepts of controllability and observability. The authors, Marius Tucsnak and George Weiss, are from different schools of thought: Tucsnak is more familiar with PDEs, while Weiss is more familiar with functional analysis. The book combines these two approaches, using functional analytic methods to formulate and investigate the main concepts. When applying these concepts to systems governed by PDEs, new difficulties arise, which require refined techniques of mathematical analysis. The book includes many examples from PDEs, with detailed worked-out solutions. It also references some more advanced results that require advanced tools from functional analysis or PDEs. The authors acknowledge the contributions of several colleagues and collaborators, including Birgit Jacob, who provided results on exact observability for normal semigroups. The book is supported by appendices that provide background on functional analysis, Sobolev spaces, differential calculus, and unique continuation for elliptic operators.This book provides an elementary introduction to the theory of observation and control for operator semigroups, focusing on admissibility, observability, and controllability. It is intended for readers with no prior knowledge of finite-dimensional control theory or operator semigroups. The book assumes a basic understanding of bounded operators on Hilbert spaces, differential equations, Fourier and Laplace transforms, distributions, and Sobolev spaces. Much of the necessary background is provided in the appendices. The book discusses the evolution of systems modeled by linear partial differential equations (PDEs) or linear delay differential equations, which can be described by operator semigroups. The state of such systems is an element in an infinite-dimensional normed space. The main topics include admissibility, observability, controllability, stabilizability, and detectability. The book covers various concepts of controllability and observability, with a focus on exact controllability and exact observability. Exact controllability is important because it guarantees stabilizability and the existence of a linear quadratic optimal control. Exact observability guarantees the existence of an exponentially converging state estimator and the existence of a linear quadratic optimal filter. The book also discusses other concepts of controllability and observability. The authors, Marius Tucsnak and George Weiss, are from different schools of thought: Tucsnak is more familiar with PDEs, while Weiss is more familiar with functional analysis. The book combines these two approaches, using functional analytic methods to formulate and investigate the main concepts. When applying these concepts to systems governed by PDEs, new difficulties arise, which require refined techniques of mathematical analysis. The book includes many examples from PDEs, with detailed worked-out solutions. It also references some more advanced results that require advanced tools from functional analysis or PDEs. The authors acknowledge the contributions of several colleagues and collaborators, including Birgit Jacob, who provided results on exact observability for normal semigroups. The book is supported by appendices that provide background on functional analysis, Sobolev spaces, differential calculus, and unique continuation for elliptic operators.