We use seismic noise cross-correlations to obtain a 3-D tomography model of SV-wave velocities beneath the western Indian Ocean, in the depth range of the oceanic crust and uppermost mantle. The study area covers 2000×2000 km2 between Madagascar and the three spreading ridges of the Indian Ocean, centred on the volcanic hotspot of La Réunion. We use seismograms from 38 ocean bottom seismometers (OBSs) deployed by the RHUM-RUM project and 10 island stations on La Réunion, Madagascar, Mauritius, Rodrigues, and Tromelin. Phase cross-correlations are calculated for 1119 OBS-to-OBS, land-to-OBS, and land-to-land station pairs, and a phase-weighted stacking algorithm yields robust group velocity measurements in the period range of 3-50 s. We demonstrate that OBS correlations across large interstation distances of >2000 km are of sufficiently high quality for large-scale tomography of ocean basins. Many OBSs yielded similarly good group velocity measurements as land stations. Besides Rayleigh waves, the noise correlations contain a low-velocity wave type propagating at 0.8-1.5 km/s over distances exceeding 1000 km, presumably Scholte waves travelling through seafloor sediments. The 100 highest-quality group velocity curves are selected for tomographic inversion at crustal and lithospheric depths. The inversion is executed jointly with a data set of longer-period, Rayleigh-wave phase and group velocity measurements from earthquakes, which had previously yielded a 3-D model of Indian Ocean lithosphere and asthenosphere. Robust resolution tests and plausible structural findings in the upper 30 km validate the use of noise-derived OBS correlations for adding crustal structure to earthquake-derived tomography of the oceanic mantle. Relative to crustal reference model CRUST1.0, our new shear-velocity model tends to enhance both slow and fast anomalies. It reveals slow anomalies at 20 km depth beneath La Réunion, Mauritius, Rodrigues Ridge, Madagascar Rise, and beneath the Central Indian spreading ridge. These structures can clearly be associated with increased crustal thickness and/or volcanic activity. Locally thickened crust beneath La Réunion and Mauritius is probably related to magmatic underplating by the hotspot. In addition, these islands are characterised by a thickened lithosphere that may reflect the depleted, dehydrated mantle regions from which the crustal melts where sourced. Our tomography model is available as electronic supplement.