A special stochastic process, called the coalescent, is of fundamental interest in population genetics. For a large class of population models this process is the appropriate tool to analyse the ancestral structure of a sample of n individuals or genes, if the total number of individuals in the population is su$ciently large. A corresponding convergence theorem was "rst proved by Kingman in 1982 for the Wright}Fisher model and the Moran model. Generalizations to a large class of exchangeable population models and to models with overlying mutation processes followed shortly later. One speaks of the &&robustness'' of the coalescent, as this process appears in many models as the total population size tends to in"nity. This publication can be considered as an introduction to the theory of the coalescent as well as a review of the most important &&convergence-to-the-coalescent-theorems''. Convergence theorems are not only presented for the classical exchangeable haploid case but also for larger classes of population models, for example for diploid, two-sex or non-exchangeable models. A reviewlike summary of further examples and applications of convergence to the coalescent is given including the most important biological forces like mutation, recombination and selection. The general coalescent process allows for simultaneous multiple mergers of ancestral lines.