In this thesis we introduce a mixture of shifted asymmetric Laplace (SAL) distributions for model-based clustering and classification. The mixture of SAL distributions allows for the parameterization of skewness as well as location and scale. Furthermore, we extend the general SAL mixture by decomposing the component scale matrices; this results in two families of SAL mixture models and a generalization of the multivariate SAL density. In developing these models we review and utilize several facets of model-based clustering. Specifically, to estimate the parameters of our mixture models we use the well-known expectation-maximization algorithm, to choose the best fitting mixture model we consider both the Bayesian information criterion and integrated completed likelihood, and to evaluate classification performance we use the adjusted Rand index. Both simulated and real data are used to demonstrate our models.