Using the notion of remote neighborhood, we dene the separation axioms T0 and T1 in L-fuzzy topological spaces (L-fts). The relations between our denitions, Hutton and Reilly's, and Wang's are discussed, and the separations of Hutton's fuzzy unit interval and Gantner's fuzzy real line are examined. Characterizations of these L-fts are proved and a series of properties of them are investigated. Moreover, some results on minimal T0 L-fts and T1 L-fts are established, which will be used immediately in our subsequent discussion.