  We review the analog of Fontaine's theory of crystalline $p$-adic Galois
representations and their classification by weakly admissible filtered
isocrystals in the arithmetic of function fields over a finite field. There
crystalline Galois representations are replaced by the Tate modules of
so-called local shtukas. We prove that the Tate module functor is fully
faithful. In addition to this \'etale realization of a local shtuka we discuss
also the de Rham and the crystalline cohomology realizations and construct
comparison isomorphisms between these realizations. We explain how local
shtukas and these cohomology realizations arise from Drinfeld modules and
Anderson's $t$-motives. As an application we construct equi-characteristic
crystalline deformation rings, establish their rigid-analytic smoothness and
compute their dimension.
