This paper reviews a variety of results for genealogical (or line-of-descent) processes that arise in connection with the theory of some classical selectively neutral haploid population genetics models. While some new results and derivations are included, the principal aim of the paper is to demonstrate the central importance and simplicity of genealogical Markov chains in this theory. Considerable attention is given to ‘diffusion time scale’ approximations of such genealogical processes. A wide variety of results pertinent to (diffusion approximations of) the classical multi-allele single-locus WrightFisher model and its relatives are unified by this approach. Other examples where the genealogical process plays an explicit role (for example, the infinite-sites models) are discussed.