  In this work, we generalize the Cram\'er-von Mises statistic via
projection-averaging to obtain a robust test for the multivariate two-sample
problem. The proposed test is consistent against all fixed alternatives, robust
to heavy-tailed data and minimax rate optimal against a certain class of
alternatives. Our test statistic is completely free of tuning parameters and is
computationally efficient even in high dimensions. When the dimension tends to
infinity, the proposed test is shown to have comparable power to the existing
high-dimensional mean tests under certain location models. As a by-product of
our approach, we introduce a new metric called the angular distance which can
be thought of as a robust alternative to the Euclidean distance. Using the
angular distance, we connect the proposed method to the reproducing kernel
Hilbert space approach. In addition to the Cram\'er-von Mises statistic, we
demonstrate that the projection-averaging technique can be used to define
robust, multivariate tests in many other problems.
