Algebraic analysis, that is the algebraic study of systems of partial differential equations by means of module theory and homological algebra, was pioneered around 1970 by M. Kashiwara, B. Malgrange, and V.P. Palamodov. The theory of differential modules, namely modules over a noncommutative ring of differential operators, is a fashionable subject of research today. However, despite its fundamental importance in mathematics, it can only be found in specialist books and papers, and has only been applied in control theory since 1990. This book provides a self-contained and exhaustive account of algebraic analysis and its application to control systems defined by partial differential equations. The first volume presents the mathematical tools needed from both commutative algebra, homological algebra, differential geometry and differential algebra. The second volume applies these new methods in order to study the structural and input/output properties of both linear and nonlinear control systems. Hundreds of explicit examples allow the reader to gain insight and experience in these topics. The book is written at a graduate level and is intended for researchers in mathematics, mathematical physics, computer algebra, control theory, and theoretical mechanics