Kirillov Theory is a mathematical framework that connects representation theory and symplectic geometry. It primarily focuses on the study of unitary representations of Lie groups and Lie algebras, providing a way to understand how these abstract structures can be realized through linear transformations. The theory introduces the concept of orbits in the dual space of a Lie algebra, allowing mathematicians to classify representations based on geometric properties. This approach has significant implications in various fields, including quantum mechanics and mathematical physics, where understanding symmetries and their representations is crucial.