  In this paper we study the semi-stable reduction of $p$ and $p^2$-cyclic
covers of curves in equal characteristic $p>0$. The main tool we use is the
classical Artin-Schreier-Witt theory for $p^n$-cyclic covers in characteristic
$p$. Although the results of this paper concern only the cases of degree $p$
and $p^2$-cyclic cover we develop the techniques and the framework in which the
general cyclic case can be studied.
