Variable-head (slug) tests are performed in monitoring wells (MWs), flush-joint driven casings, or between packers in an open borehole. Over the past 90 years, different theories were proposed to interpret the data of overdamped (OD) slug tests. The article presents theories in historical order, and whether or not they satisfy mathematics and physics. Many detected errors could be and were fixed, but not all of them. Next, the surviving equations are examined and used to fit datasets. Statistics assess whether real progress has been made: they prove it, because the residuals decrease from the oldest equation to the most recent. The latter takes into account the ubiquitous piezometric error and the slugging dynamic effects. It is shown that it accurately represents the data of in situ OD tests. Most previous best fits did not pay attention to the data’s properties and distribution of residuals in the least-squares estimate (LSE). A key assumption of the LSE is that the fit must yield Gaussian residuals with a mean of zero. The most recent equation yields such residuals for each dataset, which was not the case with earlier models, thereby also indicating real progress. Statistical analysis of many datasets also shows that for best results, the initial water column should exceed one meter, but be lower than that producing hydraulic fracturing or separation. Many previous articles have shown that installing a MW correctly is not easy and that performing a good slug test is not easy. This article also shows that a good fit requires some control mechanisms.