We present a new semantics for a language in the family of concurrent constraint logic languages. The semantics can be based on a notion of clause unfolding, since the language is closed under this transformation. The unfolding semantics is obtained as limit of an infinite unfolding process. Unfolding is also used to define an immediate consequences operator, and, therefore, a fixpoint semantics in the typical logic programming style. The result of both the unfolding and the fixpoint semantics is a set of reactive behaviors, which are trees abstractly representing all the possible computations of a program, including deadlocks and finite failures.