This book provides an introduction to the key ideas, fundamental analyses, and efficient implementations of discontinuous Galerkin finite element methods (DG-FEM) for solving partial differential equations. It covers all important theoretical results, including an overview of relevant findings from approximation and convergence theory for numerical PDEs as well as orthogonal polynomials. Through embedded Matlab codes, the algorithms are addressed and implemented for a range of classical systems of PDEs: the Maxwell equations, the Euler equations, the incompressible Navier-Stokes equations, as well as the Poisson and Helmholtz equations.