The East African Rift system (EAR) is the archetypal continental rift and a widely proposed analogue for the early stages of evolution of passive continental margins. The three‐dimensional structure of parts of the EAR has been recently elucidated by a multifold seismic (MFS) survey of Lakes Tanganyika and Malawi (Project PROBE). Analysis of fault geometries displayed on the PROBE MFS data coupled with the more extensive 28‐kHz echosounder data has improved understanding of the geometry and kinematics of the linked fault systems that underlie the lakes. In particular, it has been recognized that profiles across (i.e., at high angles to) the rift elongation commonly display fault geometries that are not readily retrodeformable. There must therefore be significant displacement out of the plane of what would conventionally be regarded as “dip” lines, that is, perpendicular to rift elongation and parallel to the tectonic transport or extension direction. Careful determination of fault plane dip and the dip of synrift reflectors in the hanging wall demonstrates that the dip directions of the shallowest faults and the steepest hanging wall sediment dips are oriented either NW or SE. This defines the direction of maximum fault block rotation and therefore the direction of extension or tectonic transport as NW/SE, oblique to the trend of the regional rift axis and to the apparent strike of many of the major border faults in the survey area. The border faults and many other faults imaged on the PROBE MFS data must therefore have significant components of strike‐slip motion and the Tanganyika and Malawi rift zones have undergone extension oblique to and not perpendicular to their axes. Near dip‐slip normal faults, steeply dipping oblique‐slip faults and subvertical strike‐slip or transfer faults have all been recognized in a complex, linked fault system that accomplishes the extension. The overall orientation of the rift lakes and their internal segmentation are influenced by major, preexisting, subvertical fault zones within the basement. Complex accommodation zones act principally as transfer zones that allow switches in gross polarity of the border fault system. The detailed geometries of the accommodation zones result from the specific relations between juxtaposed half‐graben and the strike and internal geometry of the influencing basement structure. The variations in fault geometry, subsidence, water depth and basin or rift morphology can be better explained by an oblique‐slip extensional model influenced by basement structures than by a simple orthogonal extensional model.