This text provides an in-depth analysis of the mathematical foundations of the finite element method. It is a comprehensive reference and introduction to current research on the numerical analysis of the finite element method, as well as a working textbook for graduate courses in numerical analysis. It includes many useful figures and exercises of varying difficulty. The book should prove useful to graduate students, researchers and engineers using finite element methods. The reader should have knowledge of analysis and functional analysis, particularly Hilbert spaces, Sobolev spaces and differential calculus in normed vectors. Other than these basics, the book is mathematically self-contained.