In this paper we analyze the decision problem for fragments of first-order extensions of branching time temporal logics such as computational tree logics CTL and CTL * or Prior?s Ockhamist logic of historical necessity. On the one hand, we show that the one-variable fragments of logics like first-order CTL * — such as the product of propositional CTL * with simple propositional modal logic S5, or even the one-variable bundled first-order temporal logic with sole temporal operator ?some time in the future? — are undecidable. On the other hand, it is proved that by restricting applications of first-order quantifiers to state (i.e., path-independent) formulas, and applications of temporal operators and path quantifiers to formulas with at most one free variable, we can obtain decidable fragments. The positive decidability results can serve as a unifying framework for devising expressive and effective time-dependent knowledge representation formalisms, e.g., temporal description or spatio-temporal logics.