  We construct projective toroidal compactifications for integral models of
Shimura varieties of Hodge type. We also construct integral models of the
minimal (Satake-Baily-Borel) compactification. Our results essentially reduce
the problem to understanding the integral models themselves. As such, they
cover all previously known cases of PEL type, as well as all cases of Hodge
type involving parahoric level structures. At primes where the level is
hyperspecial, we show that our compactifications are canonical in a precise
sense. We also provide a new proof of Y. Morita's conjecture on the everywhere
good reduction of abelian varieties whose Mumford-Tate group is anisotropic
modulo center. Along the way, we demonstrate an interesting rationality
property of Hodge cycles on abelian varieties with respect to p-adic analytic
uniformizations.
