The book "Mathematical Aspects of Discontinuous Galerkin Methods" provides a comprehensive overview of the mathematical foundations and applications of Discontinuous Galerkin (dG) methods. It is directed towards graduate students and researchers in applied mathematics and numerical analysis, as well as engineers interested in understanding the mathematical underpinnings of dG methods. The book covers both linear and nonlinear problems, including first-order PDEs, diffusion equations, and systems of PDEs. It discusses the theoretical aspects of dG methods, such as well-posedness, stability, consistency, and boundedness, as well as practical implementation issues. The text also explores the relationship between dG methods and finite volume methods, and presents various numerical fluxes and limiters for solving nonlinear conservation laws. The book includes detailed analysis of convergence, error estimates, and the use of a posteriori error estimates. It also covers the application of dG methods to incompressible flows and the design of dG methods for symmetric positive systems of first-order PDEs. The book is structured into three parts, each covering different aspects of dG methods, and includes appendices on practical implementation and a bibliography of over 300 references. The text is written in a self-contained manner, making it accessible to readers with a basic understanding of PDEs and numerical analysis.The book "Mathematical Aspects of Discontinuous Galerkin Methods" provides a comprehensive overview of the mathematical foundations and applications of Discontinuous Galerkin (dG) methods. It is directed towards graduate students and researchers in applied mathematics and numerical analysis, as well as engineers interested in understanding the mathematical underpinnings of dG methods. The book covers both linear and nonlinear problems, including first-order PDEs, diffusion equations, and systems of PDEs. It discusses the theoretical aspects of dG methods, such as well-posedness, stability, consistency, and boundedness, as well as practical implementation issues. The text also explores the relationship between dG methods and finite volume methods, and presents various numerical fluxes and limiters for solving nonlinear conservation laws. The book includes detailed analysis of convergence, error estimates, and the use of a posteriori error estimates. It also covers the application of dG methods to incompressible flows and the design of dG methods for symmetric positive systems of first-order PDEs. The book is structured into three parts, each covering different aspects of dG methods, and includes appendices on practical implementation and a bibliography of over 300 references. The text is written in a self-contained manner, making it accessible to readers with a basic understanding of PDEs and numerical analysis.