[Revised entry by Roy Sorensen on April 5, 2018. Changes to: Main text, Bibliography] There is wide agreement that a term is vague to the extent that it has borderline cases. This makes the notion of a borderline case crucial in accounts of vagueness. I shall concentrate on an historical characterization of borderline cases that most commentators would accept. Vagueness will then be contrasted with ambiguity and generality. This will clarify the nature of the philosophical challenge posed by vagueness. I will then discuss some rival theories of vagueness with an emphasis on many-valued logic, supervaluationism...