New insights are given into why the problem of detecting multivariate outliers can be difficult and why the difficulty increases with the dimension of the data. Significant improvements in methods for detecting outliers are described, and extensive simulation experiments demonstrate that a hybrid method extends the practical boundaries of outlier detection capabilities. Based on simulation results and examples from the literature, the question of what levels of contamination can be detected by this algorithm as a function of dimension, computation time, sample size, contamination fraction, and distance of the contamination from the main body of data is investigated. Software to implement the methods is available from the authors and STATLIB.