  Let $G$ be a locally compact group and $\omega$ be a continuous weight on
$G$. In this paper, we first characterize bicontinuous biseparating algebra
isomorphisms between weighted $L^p$-algebras. As a result we extend previous
results of Edwards, Strichartz and Parrot on algebra isomorphisms between
$L^p$-algebras to the setting of weighted $L^p$-algebras. We then study the
automorphisms of certain weighted $L^p$-algebras on integers, applying known
results on composition operators on classical function spaces.
