  Nonparametric two sample or homogeneity testing is a decision theoretic
problem that involves identifying differences between two random variables
without making parametric assumptions about their underlying distributions. The
literature is old and rich, with a wide variety of statistics having being
intelligently designed and analyzed, both for the unidimensional and the
multivariate setting. Our contribution is to tie together many of these tests,
drawing connections between seemingly very different statistics. In this work,
our central object is the Wasserstein distance, as we form a chain of
connections from univariate methods like the Kolmogorov-Smirnov test, PP/QQ
plots and ROC/ODC curves, to multivariate tests involving energy statistics and
kernel based maximum mean discrepancy. Some connections proceed through the
construction of a \textit{smoothed} Wasserstein distance, and others through
the pursuit of a "distribution-free" Wasserstein test. Some observations in
this chain are implicit in the literature, while others seem to have not been
noticed thus far. Given nonparametric two sample testing's classical and
continued importance, we aim to provide useful connections for theorists and
practitioners familiar with one subset of methods but not others.
