In this paper the notion of extensionality is studied, for theories of of λ-calculi which arise from operational semantics. A new definition of extensionality is introduced, parameterized with respect to the particular operational semantics we want to study, and it is proved that to be extensional is equivalent to be closed under a generalized j-reduction rule. This allows to treat in an uniform way very different paradigms of computations. The examples show that this general method allows to rediscover as particular applications well known results in the literature