Content: Banach Algebras and Spectral Theory Banach Algebras: Basic Concepts Gelfand Theory Nonunital Banach Algebras The Spectral Theorem Spectral Theory of -Representations Von Neumann Algebras Notes and References Locally Compact Groups Topological Groups Haar Measure Interlude: Some Technicalities The Modular Function Convolutions Homogeneous Spaces Notes and References Basic Representation Theory Unitary Representations Representations of a Group and Its Group Algebra Functions of Positive Type Notes and References Analysis on Locally Compact Abelian Groups The Dual Group The Fourier Transform The Pontrjagin Duality Theorem Representations of Locally Compact Abelian Groups Closed Ideals in L1(G) Spectral Synthesis The Bohr Compactification Notes and References Analysis on Compact Groups Representations of Compact Groups The Peter-Weyl Theorem Fourier Analysis on Compact Groups Examples Notes and References Induced Representations The Inducing Construction The Frobenius Reciprocity Theorem Pseudomeasures and Induction in Stages Systems of Imprimitivity The Imprimitivity Theorem Introduction to the Mackey Machine Examples: The Classics More Examples, Good and Bad Notes and References Further Topics in Representation Theory The Group C Algebra The Structure of the Dual Space Tensor Products of Representations Direct Integral Decompositions The Plancherel Theorem Examples Appendices A Hilbert Space Miscellany Trace-Class and Hilbert-Schmidt Operators Tensor Products of Hilbert Spaces Vector-Valued Integrals