This book provides an introduction to pseudodifferential and Fourier integral operators, focusing on their practical applications and theoretical foundations. Volume 1 is devoted to pseudodifferential operators, while Volume 2 covers Fourier integral operators. The book is structured to present the essential aspects of these operators that have been proven useful and have stabilized in theory. It emphasizes the construction of solutions to linear partial differential equations, which serve as the foundation for the theory. The text includes detailed discussions on elliptic equations, hypoelliptic equations, and boundary problems, as well as applications to the Cauchy problem and the study of the Laplace–Beltrami operator on compact Riemannian manifolds. The book also explores the use of pseudodifferential operators in solving boundary value problems, including the Calderon operator and its properties. It discusses the analyticity of solutions up to the boundary and the application of pseudodifferential operators to subelliptic estimates. The text includes a detailed treatment of the symbolic calculus of pseudodifferential operators, their parametrices, and their role in the study of elliptic and hypoelliptic equations. Volume 2, which is not included in this summary, covers Fourier integral operators, their applications, and their role in the study of microlocal analysis and the geometry of manifolds. The book also addresses the use of Fourier integral operators in the study of the spectrum of the Laplace–Beltrami operator, including the classical estimate of eigenvalues and the Poisson formula for closed geodesics. The text is written with a focus on the $ L^2 $ space, and it provides a comprehensive overview of the theory of pseudodifferential and Fourier integral operators, including their applications in various areas of mathematics. The book is intended for readers who are familiar with the basics of real and complex analysis, functional analysis, and distribution theory. It is structured to provide a clear and systematic introduction to the subject, with a focus on the practical applications and theoretical foundations of pseudodifferential and Fourier integral operators.This book provides an introduction to pseudodifferential and Fourier integral operators, focusing on their practical applications and theoretical foundations. Volume 1 is devoted to pseudodifferential operators, while Volume 2 covers Fourier integral operators. The book is structured to present the essential aspects of these operators that have been proven useful and have stabilized in theory. It emphasizes the construction of solutions to linear partial differential equations, which serve as the foundation for the theory. The text includes detailed discussions on elliptic equations, hypoelliptic equations, and boundary problems, as well as applications to the Cauchy problem and the study of the Laplace–Beltrami operator on compact Riemannian manifolds. The book also explores the use of pseudodifferential operators in solving boundary value problems, including the Calderon operator and its properties. It discusses the analyticity of solutions up to the boundary and the application of pseudodifferential operators to subelliptic estimates. The text includes a detailed treatment of the symbolic calculus of pseudodifferential operators, their parametrices, and their role in the study of elliptic and hypoelliptic equations. Volume 2, which is not included in this summary, covers Fourier integral operators, their applications, and their role in the study of microlocal analysis and the geometry of manifolds. The book also addresses the use of Fourier integral operators in the study of the spectrum of the Laplace–Beltrami operator, including the classical estimate of eigenvalues and the Poisson formula for closed geodesics. The text is written with a focus on the $ L^2 $ space, and it provides a comprehensive overview of the theory of pseudodifferential and Fourier integral operators, including their applications in various areas of mathematics. The book is intended for readers who are familiar with the basics of real and complex analysis, functional analysis, and distribution theory. It is structured to provide a clear and systematic introduction to the subject, with a focus on the practical applications and theoretical foundations of pseudodifferential and Fourier integral operators.