Interactions between tectonic, magmatic, sedimentary and hydrothermal processes during rifting and break-up of continental lithosphere lead to a variety of rifted margin types. As potential reservoirs for mineral deposits and native hydrogen, and as sites for CO2 storage and generation of geothermal energy, rifted margins are likely to have a key role in the future transition to a carbon-neutral economy. In this Review, we discuss the wide variability of rifted margin anatomy in terms of the processes that shape them. We demonstrate that observations combined with models can provide a process-based understanding of margin evolution that allows any given region to be understood more holistically than with a static end-member type (magma-rich versus magma-poor) classification. Many margins show intermediate characteristics between those end-members. Even within end-member types, there are substantial structural variations, which are shaped by the feedbacks between inheritance, deformation, sedimentation, magmatism and fluid flow. A better understanding of these feedbacks is required to assess the potential of margins to support the carbon-neutral economy. Integration of observations and modelling will help to de-risk exploration of these environments. In particular, margins need to be characterized by integrated geophysical studies, including improved wide-angle seismic velocity models with closely spaced instruments together with advanced numerical modelling techniques.