This advanced text explores a category of mathematical problems that occur frequently in physics and other sciences. Five preliminary chapters on Hamiltonian problems and numerical methods make the book accessible to students without extensive background in this area. Chapters 6 10 explore symplectic integration, symplectic order conditions, available symplectic methods, numerical experiments, and properties of symplectic integrators. The final chapters contain more advanced material. AUTHORS: J. M. Sanz-Serna is on the faculty of Charles III University of Madrid. M. P. Calvo teaches at the University of Valladolid.