The formation, storage and chemical differentiation of magma in the Earth's crust is of fundamental importance in igneous geology and volcanology. Recent data are challenging the high-melt-fraction 'magma chamber' paradigm that has underpinned models of crustal magmatism for over a century, suggesting instead that magma is normally stored in low-melt-fraction 'mush reservoirs'(1-9). A mush reservoir comprises a porous and permeable framework of closely packed crystals with melt present in the pore space(1,10). However, many common features of crustal magmatism have not yet been explained by either the 'chamber' or 'mush reservoir' concepts(1,11). Here we show that reactive melt flow is a critical, but hitherto neglected, process in crustal mush reservoirs, caused by buoyant melt percolating upwards through, and reacting with, the crystals(10). Reactive melt flow in mush reservoirs produces the low-crystallinity, chemically differentiated (silicic) magmas that ascend to form shallower intrusions or erupt to the surface(11-13). These magmas can host much older crystals, stored at low and even sub-solidus temperatures, consistent with crystal chemistry data(6-9). Changes in local bulk composition caused by reactive melt flow, rather than large increases in temperature, produce the rapid increase in melt fraction that remobilizes these cool- or cold-stored crystals. Reactive flow can also produce bimodality in magma compositions sourced from mid- to lower-crustal reservoirs(14,15). Trace-element profiles generated by reactive flow are similar to those observed in a well studied reservoir now exposed at the surface(16). We propose that magma storage and differentiation primarily occurs by reactive melt flow in long-lived mush reservoirs, rather than by the commonly invoked process of fractional crystallization in magma chambers(14).