Summary: The concept of fuzzy set was introduced by \textit{L. A. Zadeh} [Inf. Control 8, 338-353 (1965; Zbl 0139.24606)]. This concept is considered to provide a new way for handling intrinsic ambiguity by a different viewpoint from statistical treatment. Some relations between the fuzziness and the concept of fuzzy entropy, an original description of vagueness, have been first studied by \textit{A. De Luca} and \textit{S. Termini} [ibid. 20, 301-312 (1972; Zbl 0239.94028)] and its several variations have been introduced by \textit{J. Knopfmacher} [J. Math. Anal. Appl. 49, 529-543 (1975; Zbl 0308.02061)], \textit{B. R. Ebanks} [ibid. 94, 24-37 (1983; Zbl 0523.94036)] and others. We introduce the ``fuzzy relative entropy'' expressing the relative amount of fuzziness between two fuzzy sets based on the Kullback-Leibler information in probability theory and study its properties. In paraticular, a characterization theorem of the fuzzy relative entropy is shown.