  Let $G$ be a locally compact, compactly generated group of polynomial growth
and let $\omega$ be a weight on $G$. Under proper assumptions on the weight
$\omega$, the Banach space $L^p(G,\omega)$ is a Banach \ast-algebra. In this
paper we give examples of such weighted $L^p$-algebras and we study some of
their harmonic analysis properties, such as symmetry, existence of functional
calculus, regularity, weak Wiener property, Wiener property, existence of
minimal ideals of a given hull.
