Submarine sedimentary basins parallel to the trends of passive continental margins are well explained by tensional processes between two lithospheric plates. This leaves open the enigma of many basins which strike across margins. We use examples from the North Atlantic to show that such basins may be formed during the initial rifting stage due to enhanced lithospheric thinning caused by the separation of three plates. We suggest that volcanics can be emplaced in these basins and, in the extreme case, where mantle temperature and consequently the degree of partial melting are particularly high, that basaltic flows can be thick enough to fill and hide the basins.