This text offers a systematic and practical introduction to the discontinuous finite element method. It moves from a brief review of the fundamental laws and equations governing thermal and fluid systems, through a discussion of different approaches to the formulation of discontinuous finite element solutions for boundary and initial value problems, to their applicaton in a variety of thermal-system and fluid-related problems. Mesh generation and adaptivity, parellelization algorithms and error analysis are also introduced and explained. Each chapter features worked examples and exercises illustrating situations ranging from simple benchmarks to practical engineering questions. The book forms a foundation of senior undergraduate and graduate learning and also provides scientists, applied mathematicians and research engineers with a thorough treatment of basic concepts, specific techniques and methods for the use of discontinuous Galerkin methods in fluid dynamics and heat transfer applications.