Breakthrough infections with severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) in fully vaccinated individuals are receiving intense scrutiny because of their importance in determining how long restrictions to control virus transmission will need to remain in place in highly vaccinated populations as well as in determining the need for additional vaccine doses or changes to the vaccine formulations and/or dosing intervals. Measurement of breakthrough infections is challenging outside of randomized, placebo-controlled, double-blind field trials. However, laboratory and observational studies are necessary to understand the impact of waning immunity, viral variants and other determinants of changing vaccine effectiveness against various levels of coronavirus disease 2019 (COVID-19) severity. Here, we describe the approaches being used to measure vaccine effectiveness and provide a synthesis of the burgeoning literature on the determinants of vaccine effectiveness and breakthrough rates. We argue that, rather than trying to tease apart the contributions of factors such as age, viral variants and time since vaccination, the rates of breakthrough infection are best seen as a consequence of the level of immunity at any moment in an individual, the variant to which that individual is exposed and the severity of disease being considered. We also address key open questions concerning the transition to endemicity, the potential need for altered vaccine formulations to track viral variants, the need to identify immune correlates of protection, and the public health challenges of using various tools to counter breakthrough infections, including boosters in an era of global vaccine shortages.