The primary goal of these lectures is to introduce the beginner to the finite-dimensional representations of Lie groups and Lie algebras. Intended to serve non-specialists, the concentration of the text is on examples. The general theory is developed sparingly, and then mainly as a useful and unifying language to describe phenomena already encountered in concrete cases. The book begins with a brief tour through representation theory of finite groups, with emphasis determined by what is useful for Lie Groups; in particular, the symmetric groups are treated in some detail. The focus then turns to Lie groups and Lie Algebras and finally to the heart of the course: working out the finite dimensional representations of the classical groups and exploring the related geometry. The goal of the last portion of the book is to make a bridge between the example-oriented approach of the earlier parts and the general theory.