  Let $A = \bigoplus_{n=0}^{\infty}A_n$ be a connected graded $k$-algebra over
an algebraically closed field $k$ (thus $A_0=k$). Assume that a finite abelian
group $G$, of order coprime to the characteristic of $k$, acts on $A$ by graded
automorphisms. In conjunction with a suitable cocycle this action can be used
to twist the multiplication in $A$. We study this new structure and, in
particular, we describe when properties like Artin-Schelter regularity are
preserved by such a twist. We then apply these results to examples of Rogalski
and Zhang.
