Core-complex formation driven by lithospheric extension is a first-order process of heat and mass transfer in the Earth. Core-complex structures have been recognized in the continents, at slow- and ultraslow-spreading mid-ocean ridges, and at continental rifted margins; in each of these settings, extension has driven the exhumation of deep crust and/or upper mantle. The style of extension and the magnitude of core-complex exhumation are determined fundamentally by rheology: (1) Coupling between brittle and ductile layers regulates fault patterns in the brittle layer; and (2) viscosity of the flowing layer is controlled dominantly by the synextension geotherm and the presence or absence of melt. The pressure-temperature-time-fluid-deformation history of core complexes, investigated via field- and modeling-based studies, reveals the magnitude, rate, and mechanisms of advection of heat and material from deep to shallow levels, as well as the consequences for the chemical and physical evolution of the lithosphere, including the role of core-complex development in crustal differentiation, global element cycles, and ore formation. In this review, we provide a survey of ∼40 yr of core-complex literature, discuss processes and questions relevant to the formation and evolution of core complexes in continental and oceanic settings, highlight the significance of core complexes for lithosphere dynamics, and propose a few possible directions for future research.