"Mathematics of Surfaces XIII" is a collection of technical papers that documents the results of the 13th IMA International Conference on the Mathematics of Surfaces, held in September 2009 in York, United Kingdom. This book provides a comprehensive overview of current research findings and developments in the field of surface mathematics. It includes both invited and submitted contributions that address a variety of topics, including subdivision schemes, polar patchworks, compressive algorithms for partial differential equations, and many other innovative approaches to the analysis and processing of surfaces. The types of surfaces covered range from polygonal meshes to parametric and implicit surfaces, making the book a valuable resource for researchers and practitioners in this specialized field.