A long-standing challenge in the interpretation of contractional belts is to evaluate the initial condition at onset of convergence. Integrating this initial condition and its role is a prerequisite to understand and model the structural and thermal evolution of contractional belts. This requires defining the initial condition, which corresponds to that of a rift system, and for which inheritance includes a transient thermal state and a persisting inheritance, which encompasses long-lasting structural and compositional inheritance. The aim of the presentation is to illustrate the role of rift inheritance based on the work done in the Biscay-Pyrenean system. There is a general agreement that this system documents the reactivation of an increasingly mature rift system along-strike, ranging from a mature rifted margin in the west to an immature and segmented hyperextended rift in the east. In the past, different models have been proposed to explain the preceding rifting and its influence on the subsequent reactivation. The new interpretation highlights the sequential reactivation of rift-inherited decoupling horizons and identify the specific role of exhumed mantle, hyperextended and necking domains during compressional reactivation. It also highlights the contrasting fate of rift segment centres vs. segment boundaries during convergence, explaining the non-cylindricity of internal parts of collisional orogens. The observations made in the Biscay-Pyrenean system suggest that the role of inheritance is more important during initial collision. In contrast, during later stages, the orogenic evolution is mostly controlled by the classical Coulomb Wedge theory, which may account for the simpler and more continuous architecture of external parts of collisional orogens. This may also explain why models can reproduce better the final and more external orogenic structures observed in the fold and thrust belts compared to those of initial collisional stages. The new concepts established in the Biscay-Pyrenean system are now ready to be tested at other orogenic systems that result from the reactivation of former rift systems, such as the Colombian cordilleras and the Caribbean system. The new learnings ask us to rethink existing interpretations and allow to test for potential future plays in contractional belts that are not only limited to the exploration of hydrocarbons, but also include native hydrogen or hydrothermal systems.