The south-west Barents Sea is an epicontinental platform consisting of N- to NNE-oriented basins separated by basement highs. It is underlain by thick sequences of Upper Paleozoic to Cenozoic sediments and Caledonian metamorphosed basement. The basins were formed during four distinct rift-phases in the Carboniferous, late Permian, Late Jurassic-Early Cretaceous, and Paleocene-Eocene. Progressive rifting culminated in continental breakup and seafloor spreading along the North Atlantic axis approximately 55 Ma. This study explores the structural and thermal evolution of the south-west Barents Sea. The thermal and isostatic history of basins within the south-west Barents Sea is constrained through time-forward basin modelling along two profiles, building on the earlier work of Clark et al. (2014) and the PETROBAR project. Automated basin reconstructions are applied to the two profiles using the TecMod basin modelling toolbox. The effect of continental breakup and potential near-margin underplating below the Vestbakken Volcanic Province is estimated by comparing a reference case which neglects the effect of breakup and near-margin underplating with a series of scenarios which progressively integrate both (1) continental breakup and (2) continental breakup together with magmatic underplating. Both free-air and bouger gravity anomalies are extracted from the lithospheric density distribution within a new integrated routine in TecMod. Basin models are calibrated against well data and free-air and bouger-corrected gravity anomaly data. The results imply that both breakup and underplating alter the thermal and isostatic history of sediments along the margin. The hydrocarbon potential of source rocks modelled along the margin suggest breakup has a permanent thermal effect on the present-day sediments promoted by lateral heat flow, while heat conduction by underplating is more diffuse, inhibited to some degree by deeply-buried, low-conductivity shales and limestones. The likelihood of serpentinization reactions below the deep (exceeding 15km) Bjørnøya and Tromsø basins during Jurassic-Cretaceous time is tested within all basin models, concluding that the amount of crustal stretching is too low, and syn-rift sedimentation rates too high to permit embrittlement of the entire crust and hydration of the mantle. Some recent oil and gas discoveries close to the modelled profiles are investigated within the context of the calculated maturation history, validating both the models and the extent of the petroleum systems. Some assumptions are made within the basin modelling workflow and sensitivity tests are performed to scale the uncertainty inherent in the models.