For a quarter of a century the Institute of Mathematics and Its Applications has been at the forefront of developments in the practical use of mathematics. In aerodynamics and accountancy, physics and population studies, business, industry and commerce, new and vital applications are springing up all the time. The mathematics of 'music' applies as much to train wheels as to tuning-forks, to squeeling brakes as well as violins. Other branches of the subject can help us design Underground maps, break codes or program computers. In this major book the IMA provides a comprehensive survey of these themes - and all the other recent developments in applied maths.