This introductory article provides a synopsis of our current understanding of the form and dynamics of magma reservoirs in the crust. This knowledge is based on a range of experimental, observational and theoretical approaches, some of which are multidisclipinary and pioneering. We introduce and provide a contextual background for the papers in this issue, which cover a wide range of topics, encompassing magma storage, transport, behaviour and rheology, as well as the timescales on which magma reservoirs operate. We summarize the key findings that emerged from the meeting and the challenges that remain. The study of magma reservoirs has wide implications not only for understanding geothermal and magmatic systems, but also for natural oil and gas reservoirs and for ore deposit formation. This article is part of the Theo Murphy meeting issue 'Magma reservoir architecture and dynamics'.