Mathematics is one of the oldest of the sciences, with a long and very prestigious history. The pyramid builders of Egypt, the astronomers of Babylonia, the geometers of ancient Greece, and the engineers of the early Moslem world all developed and used mathematical techniques that remain indispensable for modern science and industry. The remarkable and dramatic progress of the applied sciences over the last 400 years has been possible in part because of the rapid parallel development of mathematical methods of analysis, starting with the invention of calculus by Newton and Leibniz in the 17th century. Today, mathematics underpins almost all branches of science. The detailed modelling of weather systems, of the spread of pollutants in oil spills, of the motion of particles in high-energy accelerators, of the rates of reaction of two chemical fractions, of the statistical likelihood of a positive identification in DNA fingerprinting, of the future growth of a population of an endangered species of mammal, or of the completion time for a complex computer algorithm, of the optimal decision making in complex systems, all these and many other applications would be impossible without the sophisticated techniques of modern mathematics. In addition, there is great value in studying the foundations and methods of mathematics for their own sake, with a view to improving current applications, developing entirely new applications, or simply investigating the elegance and beauty that lies behind all mathematics.