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These proceedings collect the papers accepted for presentation at the bien- nial IMA Conference on the Mathematics of Surfaces, held in the University of Cambridge, 4-7 September 2000. While there are many international con- ferences in this fruitful borderland of mathematics, computer graphics and engineering, this is the oldest, the most frequent and the only one to concen- trate on surfaces. Contributors to this volume come from twelve different countries in Eu- rope, North America and Asia. Their contributions reflect the wide diversity of present-day applications which include modelling parts of the human body for medical purposes as well as the production of cars, aircraft and engineer- ing components. Some applications involve design or construction of surfaces by interpolating or approximating data given at points or on curves. Others consider the problem of 'reverse engineering'-giving a mathematical descrip- tion of an already constructed object. We are particularly grateful to Pamela Bye (at the Institue of Mathemat- ics and its Applications) for help in making arrangements; Stephanie Harding and Karen Barker (at Springer Verlag, London) for publishing this volume and to Kwan-Yee Kenneth Wong (Cambridge) for his heroic help with com- piling the proceedings and for dealing with numerous technicalities arising from large and numerous computer files. Following this Preface is a listing of the programme committee who with the help of their colleagues did much work in refereeing the papers for these proceedings.