The Keisler theorems dealing with the definability in first-order logic of classes of structures are adapted to atomic and molecular logics. These logics are based on Dunn's gaggle theory and generalize modal logics. We show how notions of bisimulation can be automatically defined from the truth conditions of the connectives of any atomic or molecular logic. Then, we adapt the basic notion of ultraproduct of first-order logic to our atomic and molecular logics. This allows us to state our generalized Keisler theorems. The connectives of molecular logics should be in a specific format called 'normality' for our theorems to hold. We consider modal logic, the Lambek calculus, (modal) intuitionistic logic, weakly aggregative modal logic and temporal logic as case study. In a companion article [3], we use and extend these results to first-order logic and protologics, i.e. logics such that the truth conditions of their connectives are expressible by first-order formulas.