The tectonic and structural setting in which Iceland formed was complex and unusual. Complexities still characterize Iceland and its surroundings today, with multiple ephemeral rift zones, flanking marine shallow ridges and an extraordinarily wide expanse of active volcanism normal to the plate boundary. The crust has been extensively researched but remains poorly understood. Its maximum thickness exceeds 40 km, the density of its lower part is midway between normal oceanic lower crust and mantle, and it is cold and devoid of pervasive partial melt. Beneath it the mantle low-velocity zone is anomalously thick and extends down to the base of the transition zone at similar to 650 km depth. Under this, in the lower mantle, seismic tomography studies disagree about structure. Two contrasting models for why Iceland exists - the plume model and the plate model - both enjoy significant support. The plume model is predicated on anomalously hot mantle material from the lower mantle driving volcanism in Iceland. The plate model proposes that melt volumes at Iceland are little, if any, greater than along the neighbouring marine spreading plate boundaries. In that model the great thickness of the Icelandic crust is due to large quantities of continental mid- and lower-crust emplaced via lateral ductile flow where the extensional plate boundary crosses the confluence of the Caledonian and Nagssugtoqidian orogenic belts. Which hypothesis, if either, is correct can only be settled by scientific testing. The plume model has suffered from evasion of testing because it is often assumed to be settled science, but this problem is not insoluble. Tests have already been proposed for the plate model, and some are underway. It remains for these tests to be completed and for new ones to be devised if the controversy over the origin of Iceland is to be resolved.