A variety of interventions to induce a functional cure of HIV are being explored, with the aim being to allow patients to cease antiretroviral therapy (ART) for prolonged periods of time or for life. These interventions share the goal of inducing ART-free remission from HIV pathogenesis and disease progression but achieve this in quite different ways, by reducing the size of the latent reservoir (for example, small-molecule stimulation of latently infected cells), reducing the number of target cells available for the virus (for example, gene therapy) or improving immune responses (for example, active or passive immunotherapy). Here, we consider a number of these alternative strategies for inducing post-treatment control of HIV and use mathematical modelling to predict the scale of the challenge inherent in these different approaches. For many approaches, over 99.9% efficacy will likely be required to induce durable ART-free remissions. The efficacy of individual approaches is currently far below what we predict will be necessary, and new technologies to achieve lifelong functional cure are needed.