The authors have attempted to extend the Banach fixed point theorem to fuzzy contractive mappings on different types of complete fuzzy metric spaces. They have also introduced a uniform structure on the fuzzy metric space introduced in [

*A. George* and

*P. Veeramani*, ibid. 64, No. 3, 395-399 (1994;

Zbl 0843.54014)]. The authors have proved fuzzy Banach contraction theorems in the sense of complete fuzzy metric space introduced in [

*A. George* and

*P. Veeramani*, loc. cit.] and in the sense of fuzzy completeness given by

*M. Grabiec* [ibid. 27, No. 3, 385-389 (1988;

Zbl 0664.54032)]. They have also introduced a notion called fuzzy contractive sequence and observed that every fuzzy contractive sequence is G-Cauchy (definition of Cauchy sequence as given in Grabiec) and ask the question: Is a fuzzy contractive sequence a Cauchy sequence (in this case the definition of Cauchy sequence should be taken as given in [A. George and P. Veeramani, loc. cit.].