  In this paper we recall the construction and basic properties of complex
Shimura varieties and show that these properties actually characterize them.
This characterization immediately implies the explicit form of Kazhdan's
theorem on the conjugation of Shimura varieties. As a further corollary, we
show that each Shimura variety corresponding to an adjoint group has a
canonical model over its reflex field. We also indicate how this
characterization implies the existence of a p-adic uniformization of certain
unitary Shimura varieties. In the appendix we give a complete scheme-theoretic
proof of Weil's descent theorem.
