This paper reviews the state of the art in the computation of robust estimates of multivariate location and shape using combinatorial estimators such as the minimum volume ellipsoid (MVE) and iterativeM‐and S‐estimators. We also present new results on the behavior ofM‐and S‐estimators in the presence of different types of outliers, and give the first computational evidence on compound estimators that use the MVE as a starting point for an S‐estimator. Problems with too many data points in too many dimensions cannot be handled by any available technology; however, the methods presented in this paper substantially extend the size of problem that can be successfull