Every day, density-functional theory (DFT) is routinely applied to computational modeling of molecules and materials with the expectation of high accuracy. However, in certain situations, popular density-functional approximations (DFAs) have the potential to give substantial quantitative, and even qualitative, errors. The most common class of error is delocalization error, which is an overarching term that also encompasses the one-electron self-interaction error. In our opinion, its resolution remains the greatest outstanding challenge in DFT development. In this paper, we review the history of delocalization error and provide several complimentary conceptual pictures for its interpretation, along with illustrative examples of its various manifestations. Approaches to reduce delocalization error are discussed, as is its interplay with other shortcomings of popular DFAs, including treatment of non-bonded repulsion and neglect of London dispersion. This article is categorized under: Electronic Structure Theory > Density Functional Theory.