This paper proposes a definition of a fuzzy metric space in which the distance between two points is a non-negative, upper semicontinuous, normal, convex fuzzy number. Here fuzzy numbers are as defined by

*D. Dubois* and

*H. Prade* [ibid. 2, 327-348 (1979;

Zbl 0412.03035)]. The definition includes a version of the triangle inequality analogous to that of a Menger space [cf.

*B. Schweizer* and

*A. Sklar*, Pac. J. Math. 10, 313-334 (1960;

Zbl 0091.298)] but with two bounding functions in place of the t-norm. When Max and Min are the bounding functions, the triangle inequality reduces to the classical form with addition and a partial ordering as defined by

*M. Mizumoto* and

*K. Tanaka* [in ”Advances in Fuzzy Set Theory and Applications”, M. M. Gupta, R. K. Ragade, and R. R. Yager, Eds., North-Holland, New York, 153- 164 (1979;

Zbl 0434.94026)]. The authors show that every Menger space can be regarded as a fuzzy metric space, that with a weak condition on the right bounding function a fuzzy metric space induces a metrizable uniformity on the underlying set, and that certain fixed point theorems hold.