This book is an introduction to level set methods, which are powerful numerical techniques for analysing and computing interface motion in a host of settings. They rely on a fundamental shift in how one views moving boundaries; rethinking the natural geometric Lagrangian perspective and exchanging it for an Eulerian, initial value partial differential equation perspective. The resulting numerical techniques can be used to track three-dimensional complex fronts that can develop sharp corners and change topology as they evolve. A large collection of applications is provided in the text, including examples from physics, chemistry, fluid mechanics, combustion, image processing, material science, fabrication of microelectronic components, computer vision and control theory. This book will be a useful resource for mathematicians, applied scientists, practising engineers, computer graphic artists, and anyone interested in the evolution of boundaries and interfaces.