This book introduces the theory of multigrid methods for the fast numerical solution of linear and weakly nonlinear elliptic PDE. We use the finite element method to discretize the PDE problems, as this is the most natural choice, and the reader will get a thorough treatment of finite elements. No previous exposure to numerical discretization methods is assumed. All that is required of the reader is some knowledge of matrix theory. Coding the multigrid method is difficult. This book will help the reader build basic multigrid codes using easy-to-read sample Matlab codes. We use a matrix-based approach in the first part of the book, both as a way of presenting the theory in a natural way, and as a means for translating the theory into practical codes. The operators in the text and codes have the same names, which makes reading the sample codes simple, even if the reader has never coded. We deviate from the matrix-based approach only in the presentation of the nonlinear theory in the second part, which represents an area of current research. The book takes the reader from the basics and simple implementation issues all the way to the front lines of research.