Continues with the study of fuzzy metric spaces introduced by George and Veeramani. New properties are provided and issues such as completeness, uniform continuity and fixed point theorems. 
There are new examples (some of them of particular relevance) and give results about the precompacidad in fuzzy metric spaces. Also, is the study of fuzzy metric arquimedianas and not address the issue of completeness of fuzzy metric spaces and proved that, in this aspect, there is a significant difference with the theory of metric spaces, since not all space supports fuzzy metric completion. 
Explores the notion of uniform continuity and define the concepts of property and Lebesgue equinormalidad for a fuzzy metric (similar to the classics) to demonstrate a theorem in which characterize the fuzzy metric spaces in which every real function is continuous uniformly continuous by the fact that the metric is fuzzy or equinormal satisfies the Lebesgue property. Additionally, it introduces the concept of continuous t-uniform (which has no "counterpart" in the classical theory but is closely related to the notion of contract that is provided in the last chapter) that allows the characterization of fuzzy metric spaces in which all continuous real function is t-uniformly continuous by an adequate definition of t-fuzzy metric equinormal. 
Finally it introduces the concept of implementing contractionary get fuzzy and fixed point theorems for such applications in fuzzy metric spaces. Provides that every application in a contractionary fuzzy complete fuzzy metric space in which the entire sequence is a contractionary fuzzy Cauchy sequence has a unique fixed point.