A theorem is proved that generalizes Tietze's Extension Theorem for real-valued functions to L-fuzzy continuous functions on an L-fuzzy topological space. Throughout the paper the conceptual difficulties of generalizing standard topological terms to L-fuzzy topological terms are discussed. In particular, a theory of relative topologies and relative functions for L-fuzzy topological spaces is developed. The extension of a relative L-fuzzy continuous function into the fuzzy unit interval is defined. The equivalence of L-fuzzy continuous functions and monotone families of open sets is proved. This equivalence is exploited to establish a fuzzy version of Tietze's Extension Theorem. A partial converse to the theorem is proved.