Fuzzy relations are important tools that are used in fuzzy modeling, fuzzy diagnosis, and fuzzy control, which explains why it is useful to have a good understanding of fuzzy relations and their properties. Fuzzy relations extend crisp relations just like fuzzy sets extend crisp sets. This chapter discusses the properties of relations and explains how to represent relations using matrices and directed graphs. The notions of Cartesian product, projections, and cylindrical extension of fuzzy sets are used to produce fuzzy relations from ordinary fuzzy sets. A fuzzy order relation is a fuzzy transitive relation. If a fuzzy relation is reflexive, transitive, and antisymmetric, then it is a fuzzy partial order relation. The chapter discusses the elements of fuzzy graph theory, fuzzy category theory, and fuzzy vectors. It also provides information on the applications of fuzzy relations.