Nonparametric and semiparametric mixture models are valuable tools for solving many nasty problems when a population is heterogeneous. While the maximum likelihood approach is straightforward, its computation has long been known as being difficult, if not intractable, due to the estimation of a distribution function defined on an infinite-dimensional space. In this talk, I will describe some fast algorithms that I recently developed for fitting these models; present the results of their use in several applications, including the over-dispersion problem, simultaneous hypothesis testing, the Neyman-Scott problem and mixed effects models; and discuss some implementation issues using R.