This article explores three critical topics discussed in the recent debate over concurrency (overlapping sexual partnerships): measurement of the prevalence of concurrency, mathematical modelling of concurrency and HIV epidemic dynamics, and measuring the correlation between HIV and concurrency. The focus of the article is the concurrency hypothesis – the proposition that presumed high prevalence of concurrency explains sub-Saharan Africa's exceptionally high HIV prevalence. Recent surveys using improved questionnaire design show reported concurrency ranging from 0.8% to 7.6% in the region. Even after adjusting for plausible levels of reporting errors, appropriately parameterized sexual network models of HIV epidemics do not generate sustainable epidemic trajectories (avoid epidemic extinction) at levels of concurrency found in recent surveys in sub-Saharan Africa. Efforts to support the concurrency hypothesis with a statistical correlation between HIV incidence and concurrency prevalence are not yet successful. Two decades of efforts to find evidence in support of the concurrency hypothesis have failed to build a convincing case.