Nonlinear Computational Geometry : [this IMA Volume in Mathematics and its Applications Nonlinear Computational Geometry contains papers presented at a highly successful one week workshop held on May 29 June 2 2007 on the same title] An original motivation for algebraic geometry was to understand curves and surfaces in three dimensions. Recent theoretical and technological advances in areas such as robotics computer vision computer aided geometric design and molecular biology together with the increased availability of computational resources have brought these original questions once more into the forefront of research. One particular challenge is to combine applicable methods from algebraic geometry with proven techniques from piecewise linear computational geometry (such as Voronoi diagrams and hyperplane arrangements) to develop tools for treating curved objects. These research efforts may be summarized under the term nonlinear computational geometry. This volume grew out of an IMA workshop on Nonlinear Computational Geometry in May June 2007 (organized by I.Z. Emiris R. Goldman F. Sottile T. Theobald) which gathered leading experts in this emerging field. The research and expository articles in the volume are intended to provide an overview of nonlinear computational geometry. Since the topic involves computational geometry algebraic geometry and geometric modeling the volume has contributions from all of these areas. By addressing a broad range of issues from purely theoretical and algorithmic problems to implementation and practical applications this volume conveys the spirit of the IMA workshop.