  We prove the existence of weak integral canonical models of Shimura varieties
of Hodge type in arbitrary unramified mixed characteristic $(0,p)$. As a first
application we solve a conjecture of Langlands for Shimura varieties of Hodge
type. As a second application we prove the existence of integral canonical
models of Shimura varieties of preabelian (resp. of abelian) type in mixed
characteristic $(0,p)$ with $p\Ge 3$ (resp. with $p=2$) and with respect to
hyperspecial subgroups; if $p=3$ (resp. if $p=2$) we restrict in this part I
either to the $A_n$, $C_n$, $D_n^{\dbH}$ (resp. $A_n$ and $C_n$) types or to
the $B_n$ and $D_n^{\dbR}$ (resp. $B_n$, $D_n^{\dbH}$ and $D_n^{\dbR}$) types
which have compact factors over $\dbR$ (resp. which have compact factors over
$\dbR$ in some $p$-compact sense). Though the second application is new just
for $p\Le 3$, a great part of its proof is new even for $p\Ge 5$ and corrects
[Va1, 6.4.11] in most of the cases. The second application forms progress
towards the proof of a conjecture of Milne. It also provides in arbitrary mixed
characteristic the very first examples of general nature of projective
varieties over number fields which are not embeddable into abelian varieties
and which have N\'eron models over certain local rings of rings of integers of
number fields.
