Although introduced more than 60 years ago it is only during the last 15 years that there has been a systematic development of saddlepoint approximations. These approximations give a highly accurate expression for the tail of a distribution, not only in the centre of the distribution but also for very small tail probabilities. The price for this is a more cumbersome formula, the evaluation of which sometimes requires the use of a small personal computer. This book explains the ideas behind the saddlepoint approximations as well as giving a detailed mathematical description of the subject. The emphasis is two- fold. One is on popularizing the formulae through many worked out and ready to use examples. The second is on giving a comprehensive mathematical background for further research in the field. Some of the major subjects treated are uniformity of the approximations, tests in exponential families, and compound sums with applications in insurance mathematics.