Kazhdan-Lusztig polynomials are mathematical objects that arise in the study of representation theory and geometry. They are associated with Coxeter groups and provide a way to understand the structure of Hecke algebras. These polynomials encode important information about the relationships between various representations and can be used to compute dimensions of certain vector spaces. Developed by David Kazhdan and George Lusztig in the 1970s, these polynomials have applications in diverse areas such as combinatorics, algebraic geometry, and mathematical physics. They play a crucial role in the theory of Schubert varieties and help in understanding the intersection cohomology of these geometric objects.