We investigate extensions of dependence logic with generalized quantifiers. We also introduce and investigate the notion of a generalized atom. We define a system of semantics that can accommodate variants of dependence logic, possibly extended with generalized quantifiers and generalized atoms, under the same umbrella framework. The semantics is based on pairs of teams, or double teams. We also devise a game-theoretic semantics equivalent to the double team semantics. We make use of the double team semantics by defining a logic (...) $$\hbox {DC}^2$$ DC 2 which canonically fuses together two-variable dependence logic $$\hbox {D}^2$$ D 2 and two-variable logic with counting quantifiers $$\hbox {FOC}^2$$ FOC 2 . We establish that the satisfiability and finite satisfiability problems of $$\hbox {DC}^2$$ DC 2 are complete for $$\hbox {NEXPTIME}$$ NEXPTIME. (shrink)