This chapter introduces the concept of uncertainty theory, which is a branch of mathematics that generalizes classical measure theory by relaxing the additivity axiom. The theory is based on four key axioms: normality, monotonicity, self-duality, and countable subadditivity. These axioms ensure that the measure of an event reflects its likelihood in a way that is consistent with mathematical properties. The chapter begins by discussing the limitations of classical measures, particularly the challenges posed by Choquet's theory of capacities and Sugeno's fuzzy measure theory. It then introduces the idea of "self-duality" and "countable subadditivity" as more essential than "continuity" and "semicontinuity." This leads to the development of a new uncertainty theory that weakens the additivity axiom. The chapter defines an uncertain measure as a set function that satisfies the four axioms mentioned above. It provides examples to illustrate how probability measures, credibility measures, and chance measures can be instances of uncertain measures. Additionally, it includes a detailed example to demonstrate that a set function can be an uncertain measure even if it does not satisfy the axioms of probability or credibility measures.This chapter introduces the concept of uncertainty theory, which is a branch of mathematics that generalizes classical measure theory by relaxing the additivity axiom. The theory is based on four key axioms: normality, monotonicity, self-duality, and countable subadditivity. These axioms ensure that the measure of an event reflects its likelihood in a way that is consistent with mathematical properties. The chapter begins by discussing the limitations of classical measures, particularly the challenges posed by Choquet's theory of capacities and Sugeno's fuzzy measure theory. It then introduces the idea of "self-duality" and "countable subadditivity" as more essential than "continuity" and "semicontinuity." This leads to the development of a new uncertainty theory that weakens the additivity axiom. The chapter defines an uncertain measure as a set function that satisfies the four axioms mentioned above. It provides examples to illustrate how probability measures, credibility measures, and chance measures can be instances of uncertain measures. Additionally, it includes a detailed example to demonstrate that a set function can be an uncertain measure even if it does not satisfy the axioms of probability or credibility measures.