Compressional structures of Cenozoic age are ubiquitous features of the NE Atlantic margin between the western Barents Sea and offshore western Ireland. The structural suite includes simple domes or anticlines, reverse faults and broad-scale inversions. Our analysis focuses on a recently delineated group of structures in the Norwegian Sea, although these are placed in the context of similar features on the Barents margin, West of Shetlands, on the Faroes and their surrounding shelf, and in the Rockall Trough. Some (although not all) of the compressional anticlines were formed at the sites of pre-existing Cretaceous-Palaeocene depocentres. They show a multi-phase growth history. In the Norwegian Sea, particularly important phases occurred in the middle Eocene to early Oligocene and in the Miocene. We interpret the formation of these structures as a natural outcome of the transition to sea-floor spreading that occurred in the early Eocene. From this time, extremely thick sedimentary successions that had accumulated during some 300 million years of extensional tectonics were subjected to mild compression. The overall compressive stress field can be explained in terms of spreading in the adjacent ocean (ridge push) and by the distant effects of Alpine tectonics. In a plate-wide sense these effects can be regarded as two sides of the same coin. The origin of the Norwegian Sea structures is most easily visualized in terms of ridge push. NW-SE transfer zones, characteristic of the entire margin, are strongly implicated in these tectonics. A kinematic model is described that links significant structuring with a change in the spreading direction of Oligocene-Miocene age (35-20 Ma, A13-6). The compressional structures are mainly observed by their effect on sediments and volcanics of Cretaceous and Cenozoic age. They are frequently expressed in the present-day sea-floor relief, and in the case of the Faroe islands are probably responsible for present subaerial exposure. From the point of view of hydrocarbon exploration, the Cenozoic compressive anticlines have obvious potential as fourway dip closures or as components of structural-stratigraphic traps. The NW-SE fractures, orientated parallel or sub-parallel to the maximum horizontal stress direction, were probably periodically open for fluid flow from the time of NE Atlantic opening and onwards. They may therefore have facilitated migration from deeper source rocks or remigration from pre-existing hydrocarbon accumulations.