Broadly neutralizing antibodies (bNAbs), able to prevent viral entry by diverse global viruses, are a major focus of HIV vaccine design, with data from animal studies confirming their ability to prevent HIV infection. However, traditional vaccine approaches have failed to elicit these types of antibodies. During chronic HIV infection, a subset of individuals develops bNAbs, some of which are extremely broad and potent. This review describes the immunological and virological factors leading to the development of bNAbs in such "elite neutralizers". The features, targets and developmental pathways of bNAbs from their precursors have been defined through extraordinarily detailed within-donor studies. These have enabled the identification of epitope-specific commonalities in bNAb precursors, their intermediates and Env escape patterns, providing a template for vaccine discovery. The unusual features of bNAbs, such as high levels of somatic hypermutation, and precursors with unusually short or long antigen-binding loops, present significant challenges in vaccine design. However, the use of new technologies has led to the isolation of more than 200 bNAbs, including some with genetic profiles more representative of the normal immunoglobulin repertoire, suggesting alternate and shorter pathways to breadth. The insights from these studies have been harnessed for the development of optimized immunogens, novel vaccine regimens and improved delivery schedules, which are providing encouraging data that an HIV vaccine may soon be a realistic possibility.