Let $G$ be an additive finite abelian group and $Γ\subset \operatorname{End} (G)$ be a subset of the endomorphism group of $G$. A sequence $S = g_1 \cdot \ldots \cdot g_{\ell}$ over $G$ is a ($Γ$-)weighted zero-sum sequence if there are $γ_1, \ldots, γ_{\ell} \in Γ$ such that $γ_1 (g_1) + \ldots + γ_{\ell} (g_{\ell})=0$. We construct transfer homomorphisms from norm monoids (of Galois algebraic number fields with Galois group $Γ$) and from monoids of positive integers, represented by binary quadratic forms, to monoids of weighted zero-sum sequences. Then we study algebraic and arithmetic properties of monoids of weighted zero-sum sequences.