This textbook complements but goes beyond many existing texts that introduce the concept of the finite element method and its numerous applications in virtually all engineering disciplines. The focus of this book branches out to two essential thermal science areas. These are the heat transfer and fluid flow disciplines. First, there is a total of four introductory chapters, which (combined) introduce the general finite element concept. Next, the finite element heat transfer applications are discussed. The heat transfer applications vary from the simple one-dimensional thin-rod problem all the way up to a fully three-dimensional heat-conduction problem with various boundary conditions and internal heat absorption. In the fluid mechanics part of the book, each chapter is focused on a single flow problem category varying in complexity from a simple incompressible potential flow field to the full-scale three-dimensional turbulent flow structure in complex-geometry computational domains. Complex problems such as the perturbation technique (within a finite element framework) are also addressed. Among these, the common factor is in the means with which the finite element model is uniquely adapted to overcome a historically persisting analytical challenge in this and other similar problems under the same classification. While emphasizing the problem's fluid mechanics foundation, including such aspects as the extraction of the computational domain from a larger system and the choice of sound engineering boundary conditions, an equal emphasis is placed on handling the inherent analytical challenge. Also explained in each case is the choice of a suitable finite element formulation, particularly in problems involving convection-dominated flow fields.