In the context of rift basin formation, structural inheritance describes the influence of pre-existing basement structures on new, rift-related structures, including faults. Examples of basin features influenced by inheritance include rift localisation and segmentation at the plate scale, as well as variations in the geometries, orientations, and kinematics of individual rift-related faults. Given that continental rifts commonly form in pre-deformed crust, structural inheritance is likely to be the norm, not the exception. As such, structural inheritance has implications for reconstructing the paleotectonic history of rifts, investigating seismic hazards, and understanding the fluid transport and storage capabilities of natural fracture systems in the context of geo-energy and ore deposits.Our review of the literature shows that inheritance is driven by several mechanisms, which include frictional reactivation and local re-orientation of the far-field strain and/or stress. Here we highlight how insights from field observations, geophysics, and analogue and numerical models can be used to classify these mechanisms in terms of hard-linked and soft-linked inheritance. We demonstrate how different inheritance mechanisms can produce different geometric and kinematic relationships between pre-existing basement structures and rift-related faults, and that these mechanisms can be active at different depths within the same rift. Our aim is to provide a framework for recognising various expressions of structural inheritance and their underlying mechanism(s) in natural rifts, so that we can better interpret basement structures under cover and are equipped with additional constraints for understanding the multi-stage evolution of basement-influenced rift basins worldwide.