We assess the feasibility of reaching the herd immunity threshold against SARS-CoV-2 through vaccination, considering vaccine effectiveness (VE), transmissibility of the virus and the level of pre-existing immunity in populations, as well as their age structure. If highly transmissible variants of concern become dominant in areas with low levels of naturally-acquired immunity and/or in populations with large proportions of < 15 year-olds, control of infection without non-pharmaceutical interventions may only be possible with a VE ≥ 80%, and coverage extended to children.