This article is a review of the recent debate over the concurrency hypothesis — the proposition that overlapping sexual partnerships explain sub-Saharan Africa’s extraordinary HIV epidemics. Early efforts to model HIV epidemic dynamics and sexual networks substantially overstated the importance of concurrency. Recent models, including our own, show that concurrent sexual partners cannot explain contemporary HIV epidemics in eastern and southern Africa or track the early dramatic rise in HIV prevalence during the 1980s and 1990s. Survey data, including from recent surveys, show that the prevalence of concurrency is too low to explain the growth of HIV epidemics. The prevalence of concurrency and HIV do not correlate within or across countries or regions. Recent attempts to demonstrate such a correlation using the share of outside infections among total seroconversions in stable couples draws inferences that are not valid. Participants in the recent debate over concurrency disagree over the importance of coinfections in HIV epidemics in sub-Saharan Africa. Twenty-one recently published articles address the issue of concurrency. None them provides evidence for the concurrency hypothesis and many of them seriously erode the already weak case for the hypothesis. That finding has important implications for future research and HIV-prevention and treatment policy.