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We use methods and definitions from fuzzy set theory to generalize results concerning hypergraphs and intersection graphs. For each fuzzy structure defined, we use cut-level sets to define an associated sequence of crisp structures. The primary goal is then to determine what properties of the sequence of crisp structures characterize a given property of the fuzzy structure. In Chapter (2) we pay particular attention to the family of fuzzy transversals of a fuzzy hypergraph. We give an algorithmic method to construct fuzzy transversals. We also generalize the vertex coloring lemma of Berge, providing a characterization of the family of all minimal fuzzy transversals of a fuzzy hypergraph. In Chapter (3) we use similar methods to define and characterize the family of vertex colorings of a fuzzy hypergraph. In Chapter (4) we use the max and min operators to define the fuzzy Intersection graph of a family of fuzzy sets. We show that every fuzzy graph without loops is the intersection graph of some family of fuzzy sets. We show that the Gilmore and Hoffman characterization of interval graphs extends naturally to fuzzy interval graphs, but the Fulkerson and Gross characterization does not.