The thick sedimentary cover (≤18 km) along the Eastern Continental Margin of India (ECMI) and over the Bay of Bengal has hindered understanding of the mechanisms of rifting that operated on the eastern Indian and Bangladesh margins. Analysis of multichannel seismic reflection data, together with 3D gravity inversion and 2D gravity forward modelling, illuminates basement configuration, crustal thickness and Moho topography, thereby revealing the modes of rift evolution and the location of the continent–ocean boundary (COB) along these margins. The basement, particularly in the shelf and slope regions of eastern peninsular India, is traversed by nearly coast-perpendicular graben, and their seaward continuity is delineated up to 125 km from the coastline. Three different types of continental margin are present along the ECMI: (1) sheared rift on the southern segment of the ECMI, up to 14° N latitude (offshore the Southern Granulite Terrain); (2) hyper-extended rift in the central segment between 14° N and 17° N (offshore the Dharwar Craton); and (3) hypo-extended rift on the northern segment between 17° N and 20° N (offshore the Eastern Ghats Mobile Belt). The graben terminations, crustal thickness and Moho topography of both the ECMI and offshore Bengal Basin clearly suggest that the COB runs nearly parallel to the coastline of peninsular India, but takes an orthogonal turn offshore in the Bengal Basin and connects the palaeo-continental shelf and the Rajmahal–Sylhet Line onshore in the Bengal Basin. Crustal thickness (8–10 km) and depth to the Moho discontinuity (≤28 km) beneath the Bangladesh margin are unusual because the crustal thickness increased through interaction of the Kerguelen plume with the existing oceanic lithosphere. The Moho deepened largely due to the load of the Bengal Fan sediments. The absence of rifted crustal blocks on the Bangladesh margin and the continuity of the COB into the onshore Bengal Basin together imply that the present Bangladesh region was under marine conditions at least until the beginning of Bengal Fan sedimentation (i.e. no less than 23 Ma).