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Density functional theory is the most widely used method today for electronic structure calculations because of the approach proposed by Kohn and Sham in 1965: to replace the original many-body problem by an auxiliary independent-particle problem. This is an ansatz that, in principle, leads to exact calculations of properties of many-body systems using independent-particle methods; in practice, it has made possible approximate formulations that have proved to be remarkably successful. As a self-consistent method, the Kohn–Sham approach involves independent particles but an interacting density, an appreciation of which clarifies the way the method is used. The present chapter is devoted to the basic formulation of the Kohn–Sham approach and the ideas behind the crucial ingredient, the exchange–correlation energy functional Exc[n]. Information on approximate functionals in widespread use is deferred to Ch. 8, and methods for solution of the Kohn–Sham equations using the functionals are the subjects of Ch. 9 and much of the remainder of this tome.