twelve years have passed since the publication of the first edition of a multigrid tutorial. during those years, the field of multigrid and multilevel methods has expanded at a tremendous rate, reflecting progress in the development and analysis of algorithms and in the evolution of computing environments. because of these changes, the first edition of the book has become increasingly outdated and the need for a new edition has become quite apparent. with the overwhelming growth in the subject, an area in which i have neverdone serious research, i felt remarkably unqualified to attempt a new edition. re-alizing that i needed some help, i recruited two experts to assist with the project.steve mccormick (department of applied mathematics, university of colorado at boulder) is one of the original researchers in the field of multigrid methods and the real instigator of the first edition. there could be no better collaborator on the subject. van emden henson (center for applied scientific computing, lawrence livermore national laboratory) has specialized in applications of multigrid meth-ods, with a particular emphasis on algebraic multigrid methods. our collaboration on a previous siam monograph made him an obvious choice as a co-anthor.