We initiate the study of $λ$-fold near-factorizations of groups with $λ> 1$. While $λ$-fold near-factorizations of groups with $λ= 1$ have been studied in numerous papers, this is the first detailed treatment for $λ> 1$. We establish fundamental properties of $λ$-fold near-factorizations and introduce the notion of equivalence. We prove various necessary conditions of $λ$-fold near-factorizations, including upper bounds on $λ$. We present three constructions of infinite families of $λ$-fold near-factorizations, highlighting the characterization of two subfamilies of $λ$-fold near-factorizations. We discuss a computational approach to $λ$-fold near-factorizations and tabulate computational results for abelian groups of small order.