  We present some results concerning the generalized homologies associated with
nilpotent endomorphisms $d$ such that $d^N=0$ for some integer $N\geq 2$. We
then introduce the notion of graded $q$-differential algebra and describe some
examples. In particular we construct the $q$-analog of the simplicial
differential on forms, the $q$-analog of the Hochschild differential and the
$q$-analog of the universal differential envelope of an associative unital
algebra.
