The basic ideas of the saddlepoint approximation are due to H. E. Daniels [Ann. Math. Stat. 25, 631–650 (1954; Zbl 0058.35404)] and F. R. Hampel [Proc. Prague Symp. Asympt. Stat., Vol. II, Prague 1973, 109–126 (1974; Zbl 0346.62023)], and now the saddlepoint techniques are found to be useful to derive accurate approximations to distributions of statistics. The authors explain how the saddlepoint techniques can be used to obtain asymptotic expansions which give accurate results even for small sample sizes and how to find good approximations to the density of statistics in situations where the computation of the exact density is intractable. Many numerical examples are shown.