This book presents the mathematical theory of finite elements, starting from basic results on approximation theory and finite element interpolation and building up to more recent research topics, such as subgrid viscosity methods and Discontinuous Galerkin methods. The main body of the text is organized into three main sections. The first part develops the theoretical basis for the finite element method, emphasizing inf-sup conditions over the more conventional Lax-Milgrim paradigm, while the second and third parts address various applications and practical implementaions of the method, respectively. Written at the graduate level, the text contains numerous examples and exercises and will be beneficial to students and researchers alike. Depending on one's interests, several reading paths can be followed, emphasizing either convergence results, numerical algorithms, code efficiency, or applications in the engineering sciences.Ern, Alexandre is the author of 'Theory and Practice of Finite Elements', published 2004 under ISBN 9780387205748 and ISBN 0387205748.