Population geneticists study the dynamics of alternative genetic types in a replicating population. Most theoretical works rests on a simple Markov chain, called the Wright-Fisher model, to describe how an allele's frequency changes from one generation to the next. We have introduced a broad class of Markov models that share the same mean and variance as the Wright-Fisher model, but may otherwise differ. Even though these models all have the same variance effective population size, they encode a rich diversity of alternative forms of genetic drift, with significant consequences for allele dynamics. We have characterized the behavior of standard population-genetic quantities across this family of generalized models. The generalized population models can produce startling phenomena that differ qualitatively from classical behavior -- such as assured fixation of a new mutant despite the presence of genetic drift. We have derived the forward-time continuum limits of the generalized processes, analogous to Kimura's diffusion limit of the Wright-Fisher process. Finally, we have shown that some of these exotic models are more likely than the Wright-Fisher model itself, given empirical data on genetic variation in Drosophila populations. Joint work with Ricky Der and Charlie Epstein.