As an infant learns their native language, they must learn to recognize words in a variety of different contexts: different sentences, spoken at different speeds and in different ways. The infant must learn how much variation is permissible within a single word or sound category, and what sorts of variants are most likely to occur. Computational cognitive models provide insight into the acquisition process by showing what sorts of evidence are most useful for solving these learning problems. Yet many existing models are evaluated on unrealistically non-variable speech data. This talk will present work on modeling variability at the discrete (symbolic) level as well as some preliminary investigations into models of acoustics. I will discuss non-parametric Bayesian systems which learn words and sound categories from data, and compare their predictions to experimental evidence from language acquisition.