Milne, J.S., Shimura varieties and motives, In: Motives (Eds. U. Jannsen, S. Kleiman, J.-P. Serre), Proc. Symp. Pure Math., AMS, 55, 1994, Part 2, pp. 447--523.except that the TeX has been updated, some corrections and a few minor editorial changes made, and some footnotes added. Significant changes to the original article have been noted in footnotes. The numbering is unchanged.
Deligne has expressed the hope that a Shimura variety whose weight is defined over Q is the moduli variety for a family of motives. Here we prove that this is the case for "most" Shimura varieties. As a consequence, for these Shimura varieties, we obtain an explicit interpretation of the canonical model and a modular description of its points in any field containing the reflex field. Moreover, when we assume the existence of a sufficiently good theory of motives in mixed characteristic, we are able to obtain a description of the points on the Shimura variety modulo a prime of good reduction.