Kazhdan-Lusztig theory is a branch of mathematics that studies the representation theory of semisimple Lie algebras and Hecke algebras. Developed by David Kazhdan and George Lusztig in the 1970s, it provides a framework for understanding the relationships between various algebraic structures and their representations. The theory introduces Kazhdan-Lusztig polynomials, which encode important information about the geometry of flag varieties and the combinatorial properties of Coxeter groups. These polynomials play a crucial role in the study of character sheaves and have applications in areas such as geometry, combinatorics, and mathematical physics.