  We develop the analog in equal positive characteristic of Fontaine's theory
for crystalline Galois representations of a p-adic field. In particular we
describe the analog of Fontaine's functor which assigns to a crystalline Galois
representation an isocrystal with a Hodge filtration. In equal characteristic
the role of isocrystals and Hodge filtrations is played by z-isocrystals and
Hodge-Pink structures. The latter were invented by Pink. Our first main result
in this article is the analog of the Colmez-Fontaine Theorem that "weakly
admissible implies admissible". Next we construct period spaces for Hodge-Pink
structures on a fixed z-isocrystal. These period spaces are analogs of the
Rapoport-Zink period spaces for Fontaine's filtered isocrystals in mixed
characteristic and likewise are rigid analytic spaces. For our period spaces we
prove the analog of a conjecture of Rapoport and Zink stating the existence of
a "universal local system" on a Berkovich open subspace of the period space. As
a consequence of "weakly admissible implies admissible" this Berkovich open
subspace contains every classical rigid analytic point of the period space. As
the principal tool to demonstrate these results we use the analog of Kedlaya's
Slope Filtration Theorem which we also formulate and prove here.
