Lie groups This book is intended for a one year graduate course on Lie groups. Rather than providing a comprehensive treatment the author emphasizes the beautiful representation theory of compact groups. However this book also discusses important topics such as the Bruhat decomposition and the theory of symmetric spaces. TOC:Haar Measure * Schur Orthogonality * Compact Operators * The Peter Weyl Theorem * Lie Subgroups of GL(n C) * Vector Fields * Left Invariant Vector Fields * The Exponential Map * Tensors and Universal Properties * The Universal Enveloping Algebra * Extension of Scalars * Representations of sl(2 C) * The Universal Cover * The Local Frobenius Theorem * Tori * Geodesics and Maximal Tori * Topological proof of Cartans Theorem * The Weyl Integration Formula * The Root System * Examples of Root Systems * Abstract Weyl Groups * The Fundamental Group * Semisimple Compact Groups * Highest Weight Vectors * The Weyl Character Formula * Spin * Complexification * Coxeter Groups * The Iwasawa Decomposition * The Bruhat Decomposition * Symmetric Spaces * Relative Root Systems.* Embeddings of Lie Groups * Mackey Theory * Characters of GL(n C) * Duality between Sk and GL(n C) * The Jacobi Trudi Identity * Schur Polynomials and GL(n C) * Schur Polynomials and Sk * Random Matrix Theory * Minors of Toeplitz Matrices * Branching Formulae and Tableaux * The Cauchy Identity * Unitary branching rules * The Involution Model for Sk * Some Symmetric Algebras * Gelfand Pairs * Hecke Algebras * Cohomology of Grassmannians