The innovative concept of complex fuzzy sets is introduced. The novelty of the complex fuzzy set lies in the range of values its membership function may attain. In contrast to a traditional fuzzy membership function, this range is not limited to [0,1] but extended to the unit circle in the complex plane. Thus, the complex fuzzy set provides a mathematical framework for describing membership in a set in terms of a complex number. A study of this original concept is presented, including examples of possible applications, which demonstrate the new theory.