Abstract Team semantics is the mathematical framework of modern logics of dependence and independence in which formulae are interpreted by sets of assignments (teams) instead of single assignments as in first-order logic. In order to deepen the fruitful interplay between team semantics and database dependency theory, we define Polyteam Semantics in which formulae are evaluated over a family of teams. We begin by defining a novel polyteam variant of dependence atoms and give a finite axiomatization for the associated implication problem. We relate polyteam semantics to team semantics and investigate in which cases logics over the former can be simulated by logics over the latter. We also characterize the expressive power of poly-dependence logic by properties of polyteams that are downwards closed and definable in existential second-order logic ($\textsf{ESO}$). The analogous result is shown to hold for poly-independence logic and all $\textsf{ESO}$-definable properties. We also relate poly-inclusion logic to greatest fixed point logic.