The dark sector offers a compelling theoretical framework for addressing the nature of dark matter while potentially solving other fundamental problems in physics. This review focuses on light dark-flavored sector models, in which the flavor structure of the interactions with the Standard Model is nontrivial and distinguishes among different fermion families. Such scenarios feature flavor violation that leads to unique experimental signatures, such as flavor-changing neutral current decays of heavy hadrons (kaons, D and B mesons, baryons) and leptons (muons and taus) with missing energy carried away by light dark-sector particles. In this article, we review their motivation, summarize current constraints, highlight discovery opportunities in ongoing and future flavor experiments, and discuss implications for astrophysics and cosmology.