  In this paper we introduce the classical and quantum covariant Weil algebras.
Covariant Weil algebras are simultaneous generalizations of Weil algebras and
family algebras. We will define differentials, Lie derivatives and contractions
on them to make them curved-dg algebras. Moreover, the expression of curvatures
will be given. It is hoped that covariant Weil algebras can be used in the
construction of Mackey's analogue.
