  We prove the Demailly--Peternell--Schneider conjecture in positive
characteristic: if $X$ is a smooth projective variety over an algebraically
closed field of characteristic $p>0$ with $-K_X$ is nef, then the Albanese
morphism $a: X \to A$ is surjective. We also show strengthenings either
allowing mild singularities for $X$, or proving more special properties of $a$.
  The above statement for compact K\"ahler manifolds was conjectured originally
by Demailly, Peternell and Schneider in 1993, and for smooth projective
varieties of characteristic zero it was shown by Zhang in 1996. In positive
characteristic, all earlier results involved tameness assumptions either on
cohomology or on the singularities of the general fibers of $a$. The main
feature of the present article is the development of a technology to avoid such
assumptions.
