A variety of results for genealogical and line-of-descent processes that arise in connection with the theory of some classical selectively neutral population genetics models are reviewed. While some new results and derivations are included, the principle aim 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 multiallele single-locus Wright-Fisher model and its relatives are simplified and unified by this approach. Other examples where such genealogical processes play an explicit role, such as the infinite sites and infinite alleles models, are discussed.