  We study isogenies between K3 surfaces in positive characteristic. Our main
result is a characterization of K3 surfaces isogenous to a given K3 surface $X$
in terms of certain integral sublattices of the second rational $\ell$-adic and
crystalline cohomology groups of $X$. This is a positive characteristic analog
of a result of Huybrechts, and extends results of the second author. We give
applications to the reduction types of K3 surfaces and to the surjectivity of
the period morphism. To prove these results we describe a theory of B-fields
and Mukai lattices in positive characteristic, which may be of independent
interest. We also prove some results on lifting twisted Fourier--Mukai
equivalences to characteristic 0, generalizing results of Lieblich and Olsson.
