  Let $G$ be a semisimple algebraic group over a field of characteristic $p >
0$. We prove that the dual Weyl modules for $G$ all have $p$-filtrations when
$p$ is not too small. Moreover, we give applications of this theorem to
$p^n$-filtrations for $n > 1$, to modules containing the Steinberg module as a
tensor factor, and to the Donkin conjecture on modules having $p$-filtrations.
