This text provides an introduction to the numerical solution of initial and boundary value problems in ordinary differential equations on a firm theoretical basis. This book strictly presents numerical analysis as a part of the more general field of scientific computing. Important algorithmic concepts are explained down to questions of software implementation. For initial value problems, a dynamical systems approach is used to develop Runge-Kutta, extrapolation, and multistep methods. For boundary value problems including optimal control problems, both multiple shooting and collocation methods are worked out in detail. Graduate students and researchers in mathematics, computer science, and engineering will find this book useful. Chapter summaries, detailed illustrations, and exercises are contained throughout the book with many interesting applications taken from a rich variety of areas. Peter Deuflhard is founder and president of the Zuse Institute Berlin (ZIB) and full professor of scientific computing at the Free University of Berlin, Department of Mathematics and Computer Science. Folkmar Bornemann is full professor of scientific computing at the Center of Mathematical Sciences, Technical University of Munich. This book was translated by Werner Rheinboldt, professor emeritus of numerical analysis and scientific computing at the Department of Mathematics, University of Pittsburgh.