The aim of this paper is to introduce new class of Fuzzy set, namely, fuzzy *g-closed set in Fuzzy topological spaces (fts). This new class is properly lies between the classes of g-closed and g*-closed fuzzy sets. Compared the fuzzy *g - closed set with already existing various fuzzy closed sets (introduced and studied by many researchers) in fuzzy topological paces. We studied the properties of obtained closed fuzzy set with necessary counter examples. Further work leads to study and analyses of the concept like, fuzzy *g-continuous, fuzzy *g -irresolute mapping, fuzzy *g -closed maps, fuzzy *g -open maps and fuzzy *g -homeomorphism mapping in Fuzzy topological spaces.