Let $X$ be a smooth variety (e.g., affine space) over a finite field (e.g., the integers modulo a prime). In the course of proving the last of Weil’s conjectures on zeta functions of varieties over finite field, Deligne studied a certain category of representations of the fundamental group of $X$ which carry information about these zeta functions. He also made a far-reaching conjecture to the effect that such objects always look as if they “come from geometry”. We will state the conjecture, describe some of its more concrete consequences, and discuss some results of various authors (L. Lafforgue, V. Lafforgue, Deligne, Drinfeld, T. Abe, Abe-Esnault, and the speaker) which very recently have led to a resolution of this 40-year-old open problem.