SUMMARY This paper is the first in a series of three (this issue) which present the results of the RAMESSES study (Reykjanes Axial Melt Experiment: Structural Synthesis from Electromagnetics and Seismics). RAMESSES was an integrated geophysical study which was carefully targeted on a magmatically active, axial volcanic ridge (AVR) segment of the Reykjanes Ridge, centred on 57°45aeN. It consisted of three major components: wide-angle seismic profiles along and across the AVR, using ocean-bottom seismometers, together with coincident seismic reflection profiles; controlled-source electromagnetic sounding (CSEM); and magnetotelluric sounding (MT). Supplementary data sets included swath bathymetry, gravity and magnetics. Analyses of the major components of the experiment show clearly that the sub-axial geophysical structure is dominated by the presence and distribution of aqueous and magmatic fluids. The AVR is underlain by a significant crustal magma body, at a depth of 2.5 km below the sea surface. The magma body is characterized by low seismic velocities constrained by the wide-angle seismic data; a seismic reflection from its upper surface; and a region of anomalously low electrical resistivity constrained by the CSEM data. It includes a thin, ribbon-like melt lens at the top of the body and a much larger region containing at least 20 per cent melt in a largely crystalline mush zone, which flanks and underlies the melt lens. RAMESSES is the first experiment to provide convincing evidence of a significant magma body beneath a slow spreading ridge. The result provides strong support for a model of crustal accretion at slow spreading rates in which magma chambers similar to those at intermediate and fast spreading ridges play a key role in crustal accretion, but are short-lived rather than steady-state features. The magma body can exist for only a small proportion of a tectono-magmatic cycle, which controls crustal accretion, and has a period of at least 20 000 years. These findings have major implications for the temporal patterns of generation and migration of basaltic melt in the mantle, and of its delivery into the crust, beneath slow-spreading mid-ocean ridges.