Abstract The mild compressional structures of Cenozoic age on the passive margins bordering Norway, the UK, the Faroes and Ireland have been the subject of much discussion in the literature. Nevertheless, their origin remains enigmatic. Candidate mechanisms must be able to explain the generation of sufficient stress to cause deformation, the episodic nature of the structures and why they developed where they did. We examine these mechanisms and conclude that multiple causes are probable, while favouring body force as potentially the most important agent. The geometry and setting of the structures are incompatible with gravitational sliding and toe-thrusting, probably the commonest ‘compressive’ structuring around the Atlantic margins. A passive mode of origin featuring drape or flank sedimentary loading probably emphasized some of the structures, but cannot be invoked as a primary mechanism. Likewise, reactivation of basement structure probably focused deformation but did not initiate it. Far-field orogenic stress from Alpine orogenic phases and from the West Spitsbergen–Eurekan folding and thrusting is also examined. This mechanism is attractive because of its potential to explain episodicity of the compressional structures. However, difficulties exist with stress transmission pathways from these fold belts, and the passive margin structures developed for much of their existence in the absence of any nearby contemporaneous orogeny. Breakup and plate spreading forces such as divergent asthenosheric flow have potential to explain early post-breakup compressional structuring, for example on the UK–Faroes margin, but are unlikely to account for later (Neogene) deformation. Ridge push, generally thought to be the dominant body force acting on passive margins, can in some circumstances generate enough stress to cause mild deformation, but appears to have low potential to explain episodicity. It is proposed here that the primary agent generating the body force was development of the Iceland Insular Margin, the significant bathymetric-topographic high around Iceland. Circumstantially, in Miocene times, this development may also have coincided with the acme of the compressional structures. We show that, dependent on the degree of lithosphere–asthenosphere coupling, the Iceland Plateau may have generated enough horizontal stress to deform adjacent margins, and may explain the arcuate distribution of the compressional structures around Iceland. Assuming transmission of stress through the basement we argue that, through time, the structures will have developed preferentially where the basement is hotter, weaker and therefore more prone to shearing at the relatively low stress levels. This situation is most likely at the stretched and most thermally-blanketed crust under the thickest parts of the young (Cretaceous–Cenozoic) basins. Although several elements of this model remain to be tested, it has the potential to provide a general explanation for passive margin compression at comparatively low stress levels and in the absence of nearby orogeny or gravitational sliding.