In this chapter we discuss two facets of uncertainty, namely, fuzziness and non-Specificity, that may be associated with fuzzy sets. After presenting a set of desirable axioms that a measure of fuzziness should satisfy various attempts to quantify fuzziness are discussed. In this regard, the need for and quantification of higher order measures of fuzziness and weighted fuzziness are also presented. Finally, we discuss a few attempts to quantify non-specificity.