The aim of this volume is to present research workers and graduate students in numerical analysis with a state-of-the-art survey of some of the most active areas of numerical analysis. This, and a companion volume, arise from a Summer School intended to cover recent trends in the subject. The chapters are written by the main lecturers at the School. Each is an internationally renowned expert in his respective field. This volume covers research in the numerical analysis of nonlinear phenomena: evolution equations, free boundary problems, spectral methods, and numerical methods for dynamical systems, nonlinear stability, and differential equations on manifolds.