A Lie group is a mathematical structure that combines algebraic and geometric concepts. It consists of a set of elements that can be continuously transformed into one another, allowing for smooth operations. Lie groups are often used to study symmetries in various fields, including physics and engineering. These groups are named after the mathematician Sophus Lie, who developed the theory in the 19th century. Lie groups can be classified into different types, such as compact and non-compact, depending on their properties. They play a crucial role in areas like differential equations and quantum mechanics.