The theory of Lie groups studies continuous symmetries and their mathematical structures. A Lie group is a group that is also a smooth manifold, allowing for the application of calculus. These groups are essential in various fields, including physics, where they describe symmetries in quantum mechanics and general relativity. Lie groups are characterized by their associated Lie algebras, which capture the group's infinitesimal structure. The relationship between Lie groups and Lie algebras helps in understanding complex systems and solving differential equations. This theory has profound implications in both mathematics and theoretical physics.