Statistical distances, divergences, and similar quantities have a large history and play a fundamental role in statistics, machine learning and associated scientific disciplines. However, within the statistical literature, this extensive role has too often been played out behind the scenes, with other aspects of the statistical problems being viewed as more central, more interesting, or more important. The behind the scenes role of statistical distances shows up in estimation, where we often use estimators based on minimizing a distance, explicitly or implicitly, but rarely studying how the properties of a distance determine the properties of the estimators. Distances are also prominent in goodness-of-fit, but the usual question we ask is "how powerful is this method against a set of interesting alternatives" not "what aspect of the distance between the hypothetical model and the alternative are we measuring?" Our focus is on describing the statistical properties of some of the distance measures we have found to be most important and most visible. We illustrate the robust nature of Neyman's chi-squared and the non-robust nature of Pearson's chi-squared statistics and discuss the concept of discretization robustness.