In this paper, we present a conformance testing theory for Barbosa's abstract components. This is made possible by defining fi rst a trace model for components by causal transfer functions, that is functions of data flow transformations rythmitical by discrete instants. This then allows us to exploit a particular analysis technique to define a test selection strategy based on test purposes de ned as some subtrees of the execution tree built from component traces. Moreover, we show in this paper that Barbosa's de nition of components is abstract enough to subsume a large family of state-base formalisms such as Mealy automata, Labeled Transition Systems (LTS), Input Output Labeled Transition Systems (IOLTS), etc. by instantiating the monads underlying Barbosa's defi nition. Hence, the conformance theory presented in this paper is de facto a generalization of standard ones we nd for di erent state-base formalisms.