The Southwest Indian Ridge represents a slow-spreading end-member of the global mid-ocean ridge system, and as such its structure places important constraints on models of melt supply and delivery from the mantle at ridges. We present results from a wide-angle seismic experiment conducted at the ridge axis at 66°E, in a region that has comprehensive swath bathymetric, gravity and magnetic data coverage and where the full spreading rate is ∼12 mm yr−1. Based on these data, the experiment traversed four spreading segments. Crustal thickness and velocity structure were determined along three intersecting profiles each ∼100 km long using shots from a 10-gun, 71 L tuned airgun array towed at 15 m depth and fired at 40 s intervals, recorded on three ocean bottom hydrophones on each profile. OBH data show high-amplitude arrivals from oceanic Layer 2, lower-amplitude arrivals from Layer 3 and wide-angle reflections from the Moho. Forward modelling and inversion of traveltime picks from these data show that the crust consists of a 1.5–2.5-km-thick Layer 2 with a high velocity gradient and a 0.5–3.0-km-thick Layer 3 with a low velocity gradient, and a crustal thickness of 2.2–5.4 km. Additional constraints on the models come from 2-D modelling of gravity data along the profiles, corrected for 3-D effects of off-line bathymetry. Along-axis, the thickness of Layer 2 varies little, but Layer 3 is thick at segment centres and very thin at segment boundaries. Along a flowline profile, crustal thickness varies by up to 75 per cent from its minimum value in ∼3 Myr. The reduced crustal thickness is consistent with observations from very slow-spreading ridge axes elsewhere and may be explained by conductive cooling of the upwelling mantle. The large along-axis variations in Layer 3 thickness indicate that magmatic accretion is focused at segment centres and melt is delivered to segment ends perhaps only by lateral dyke propagation. Flowline variations in crustal thickness may result from episodicity of melt supply on timescales of ∼3 Myr and by tectonic extension during amagmatic periods. Velocities at the top of Layer 2 are poorly correlated with crustal age based on magnetic anomalies, suggesting also that episodicity is decoupled between adjacent segments.