The Lie Algebras have now become essential to many parts of mathematics, and theoretical physics, or quantum chemistry. This course will offer an introduction to this important field. We will start with the definitions of Lie algebras, like solvable, nilpotent, simple and semi-simple Lie algebras, etc., and their characterizations. We then proceed to classify all finite-dimensional simple Lie algebras, introducing the root systems and Dynkin diagrams. The typical classical Lie algebras, and exceptional Lie algebras will be introduced. Their representations by boson fields, or Fermion fields, or differential operators give different versions of the representation theory of these algebras. In the discussions of the representation theory, the theory of finite group representations will also be introduced. If the time permits, we will introduce the Kac-Moody algebras and the Virasoro algebras as examples of infinite-dimensional algebras.