This second edition of the popular A Multigrid Tutorial preserves the introductory spirit of the first edition while roughly doubling the amount of material covered. The topics of the first edition have been enhanced with additional discussion, new numerical experiments, and improved figures. New topics in the second edition include nonlinear equations, Neumann boundary conditions, variable mesh and variable coefficient problems, anisotropic problems, algebraic multigrid (AMG), adaptive methods, ..and finite elements. This introductory book is ideally suited as a companion textbook for graduate numerical analysis courses. It is written for computational mathematicians, engineers, and other scientists interested in learning about multigrid.