  We introduce and study "norm-multiplicative" homomorphisms $\varphi: {\cal
L}^1(F) \rightarrow {\cal M}_r(G)$ between group and measure algebras, and
$\varphi: {\cal L}^1(\omega_F) \rightarrow {\cal M}(\omega_G)$ between Beurling
group and measure algebras, where $F$ and $G$ are locally compact groups with
continuous weights $\omega_F$ and $\omega_G$. Through a unified approach we
recover, and sometimes strengthen, many of the main known results concerning
homomorphisms and isomorphisms between these (Beurling) group and measure
algebras. We provide a first description of all positive homomorphisms
$\varphi: {\cal L}^1(F) \rightarrow {\cal M}_r(G)$. We state versions of our
results that describe a variety of (possibly unbounded) homomorphisms $\varphi:
\mathbb{C} F \rightarrow \mathbb{C} G$ for (discrete) groups $F$ and $G$.
