This text is intended for the beginning graduate student with minimal preparation. However since Lie groups abstract the analytic properties of matrix groups, the student is expected to have some knowledge of senior level algebra, topology, and analysis as given in юте of the references. In Chapter 1 we review some advanced calculus and extend these results to manifolds in Chapter 2. Consequently the reader knowing these results can skip these chapters but should pay attention to the examples on matrix groups. After this the reader probably should follow the order given in the contents noting that the first part of the text is about Lie groups while the algebraic study of Lie algebras begins in Chapter 9. We have not attempted to prove all basic results so the serious student should take the indicated detours to such texts as those by Freudenthal and de Vries, Helgason, Jacobson, or Wolf. In particular the student must develop his own taste in this subject and ours is only one point of view.