This book is an expanded version of the lectures given at the Nankai Mathematical Summer School in 1997. It provides an introduction to Lie groups, Lie algebras and their representations as well as a flavor of current research for graduate students who have no previous specialized knowledge of representation theory. It only assumes that the reader has a good knowledge of undergraduate linear algebra.Parts I-III of the book provide relatively elementary material about representation theory on finite groups and on compact Lie groups. A chapter on simple groups of the Lie type has been added to the original notes so that the reader can have a complete picture of simple finite groups. Part IV presents some advanced material on infinite-dimensional representations of non-compact groups. The last part of the notes touches on many important ideas of current research, such as orbit methods, minimal representations and dual pair correspondences, which may serve as an introduction to the current research on representation theory.