This doctoral level lecture course is intended to audience interested in theoretical physics and mathematics. Its purpose is to introduce the theory of semisimple Lie algebras so that the student could master their Cartan-Weyl basis, as well as to become familiar with important basic structures such as the root and weight systems, which are needed for constructing finite-dimensional irreducible representations. The final goal is the construction of the graded Lie algebras and the related Kac-Moody algebras. These are basic tools in contemporary theoretical and mathematical physics. They are fundamental tools for the infinite-dimensional completely integrable Hamiltonian systems and to a number of problems in quantum mechanics, statistical physics and others.