This graduate-level course will introduce the students to a wide varietyof topics in Representation theory, with a particular focus on representationsof Lie algebras. In the beginning of the course, introduction to Representationtheory will be given, the students will learn the basic constructions andclassical results such as Schur's lemma, Maschke's theorem, Schur— Weylduality, density theorem, Wedderburn—Artin theorem, representations of quiversand Gabriel's theorem. We then proceed to representations of Lie algebras. Thestudents will learn the following topics: representations of nilpotent andsolvable Lie algebras, finite-dimensional representations of semisimple Liealgebras, Weyl character formula, Verma modules theory and category O. Thestudents will be exposed to current research topics in Lie theory and related topics.