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The emerging concept of a magma reservoir is one in which regions containing melt extend from the source of magma generation to the surface. The reservoir may contain regions of very low fraction intergranular melt, partially molten rock (mush) and melt lenses (or magma chambers) containing high melt fraction eruptible magma, as well as pockets of exsolved magmatic fluids. The various parts of the system may be separated by a sub-solidus rock or be connected and continuous. Magma reservoirs and their wall rocks span a vast array of rheological properties, covering as much as 25 orders of magnitude from high viscosity, sub-solidus crustal rocks to magmatic fluids. Time scales of processes within magma reservoirs range from very slow melt and fluid segregation within mush and magma chambers and deformation of surrounding host rocks to very rapid development of magma and fluid instability, transport and eruption. Developing a comprehensive model of these systems is a grand challenge that will require close collaboration between modellers, geophysicists, geochemists, geologists, volcanologists and petrologists. This article is part of the Theo Murphy meeting issue ‘Magma reservoir architecture and dynamics’.