  We prove that Shimura varieties admit integral canonical models for
sufficiently large primes. In the case of abelian-type Shimura varieties, this
recovers work of Kisin-Kottwitz for sufficiently large primes. We also prove
the existence of integral canonical models for images of period maps
corresponding to geometric families. We deduce several consequences from this,
including an unramified rigid analogue of Borel's extension theorem, a version
of Tate semisimplicity, CM lifting theorems, and a weakened version of Tate's
isogeny theorem for ordinary points.
