Nonparametric two sample or homogeneity testing is a decision theoretic\nproblem that involves identifying differences between two random variables\nwithout making parametric assumptions about their underlying distributions. The\nliterature is old and rich, with a wide variety of statistics having being\nintelligently designed and analyzed, both for the unidimensional and the\nmultivariate setting. Our contribution is to tie together many of these tests,\ndrawing connections between seemingly very different statistics. In this work,\nour central object is the Wasserstein distance, as we form a chain of\nconnections from univariate methods like the Kolmogorov-Smirnov test, PP/QQ\nplots and ROC/ODC curves, to multivariate tests involving energy statistics and\nkernel based maximum mean discrepancy. Some connections proceed through the\nconstruction of a \\textit{smoothed} Wasserstein distance, and others through\nthe pursuit of a "distribution-free" Wasserstein test. Some observations in\nthis chain are implicit in the literature, while others seem to have not been\nnoticed thus far. Given nonparametric two sample testing's classical and\ncontinued importance, we aim to provide useful connections for theorists and\npractitioners familiar with one subset of methods but not others.\n