We study Fourier-Mukai equivalence of K3 surfaces in positive characteristic and show that the ical results over the complex numbers all generalize. The key result is a positive-characteristic version of the Torelli theorem that uses the derived category in place of the Hodge structure on singular cohomology ; this is proven by algebraizing formal lifts of Fourier-Mukai kernels to characteristic zero. As a consequence, any Shioda-supersingular K3 surface is uniquely determined up to isomorphism by its derived category of coherent sheaves. We also study different realizations of Mukai's Hodge structure in algebraic cohomology theories (étale, crystalline, de Rham) and use these to prove : 1) the zeta function of a K3 surface is a derived invariant (discovered independently by Huybrechts) ; 2) the variational crystalline Hodge conjecture holds for correspondences arising from Fourier-Mukai kernels on products of two K3 surfaces.