This thesis concerns select methods related to multivariate nonparametric data description, especially multivariate location. It presents and provides implementations of algorithms for computing the projection median both exactly (in low dimensions) and approximately (for use in higher dimensions). The algorithms use techniques from computational geometry and Monte Carlo methods. Further, an intuitive notion of data depth based on an average univariate ranking of points is introduced. This depth measure is shown to be quickly computable in low dimensions and easily approximated in high dimensions via Monte Carlo techniques. In addition, its theoretical properties are investigated. Several applications of these methods are demonstrated, using both real and simulated data.