This report is part of a series whose aim is to present in a synthetic way the major methods of ``analytic combinatorics'' needed in the average--case analysis of algorithms. It reviews the use of the saddle point method in order to estimate asymptotically coefficients of fast growing generating functions. The applications treated concern the enumeration of set partitions and integer partitions, permutations with cycle restrictions, increasing subsequences, as well as distribution estimates when large powers are involved.