'Nifio, nifio-dijo con voz alta a esta saz6n don Quijote-seguid vues tra historia en lfnea recta, y no os metB. is en las curvas y transver sales. ' M. de Cervantes, El Ingenioso Hidalgo Don Quijote de la Man cha, Parte II, Capltulo XXVI. 'Pray don't trouble yourself to say it any Ionger than that. ' L. Carroll, Alice's Adventures in Wonderland, Chapter IX. Recent years have witnessed a dramatic growth of the Iiterature on symplectic integration of Hamiltonian problems. While the sub ject is still changing rapidly and important discoveries may yet be made, we feel it is time to present a unified view of this interdisci plinary field. The purpose of this book is to offer such a unified first introduction. Being exhaustive in the topics included and saying the last word on every issue treated have not been amongst our aims. Some readers may be interested in integrating the Hamiltonian problems they find in their own scientific field. This sort of reader cannot reasonably be expected to be an expert in numerical meth ods. On the other hand, readers with an expertise in numerical methods may wish to enter the Hamiltonian field in order to de sign and analyse new Hamiltonian integrators. In our experience, readers in this second group are likely to be uncomfortable with the basic ideas of the Hamiltonian formalism.