Discontinous Galerkin (DG) methods for solving partial differential equations, developed in the late 1990s, have become popular among computational scientists. Covering both theory and computation, this book focuses on three primal DG methods - the symmetric interior penalty Galerkin, incomplete interior penalty Galerkin, and nonsymmetric interior penalty Galerkin - which are variations of interior penalty methods. The author provides the basic tools for analysis and discusses coding issues, including data structure, construction of local matrices, and assembling of the global matrix. Computational examples and applications to important engineering problems are also included. Part I focuses on the application of DG methods to second order elliptic problems in one dimension and in higher dimensions. Part II presents the time-dependent parabolic problems - with and without convection. Part III contains applications of DG methods to solid mechanics (linear elasticity), fluid dynamics (Stokes and Navier-Stokes), and porous media flow (two-phase and miscible displacement).