1. The Concept of a Spherical Function.- § 1.1. Review of Some Basic Notions of Representation Theory.- § 1.2. Decomposition of a Representation with Respect to a Compact Subgroup K and K-finite Representations.- § 1.3. Elementary Spherical Functions of Arbitrary Type.- § 1.4. Spherical Functions on Lie Groups.- § 1.5. Gel'fand Pairs (G, K).- § 1.6. Plancherel Formula for G/K.- § 1.7. Eigenfunction Expansions in G/K.- Notes on Chapter 1.- 2. Structure of Semisimple Lie Groups and Differential Operators on Them.- § 2.1. Groups of Class ?.- § 2.2. Iwasawa Decomposition. Roots. Weyl Group.- § 2.3. Parabolic Subalgebras and Parabolic Subgroups.- § 2.4. Integral Formulae.- § 2.5. Flag Manifolds, Bruhat Decomposition and Related Integral Formulae.- § 2.6. Differential Operators on G and G/K.- Notes on Chapter 2.- 3. The Elementary Spherical Functions.- § 3.1. Principal Series Representations and Integral Representations for Their Matrix Coefficients.- § 3.2. Determination of All Elementary Spherical Functions. The Functional Equations.- § 3.3. The Harish-Chandra Transform.- § 3.4. Finite Dimensional Representation Theory of G and Its Consequences for the H-Function and the Elementary Spherical Functions.- § 3.5. Convexity Properties of the H-Function.- Notes on Chapter 3.- 4. The Harish-Chandra Series for ?? and the c-Function.- §4.1. Radial Components of Spherical Differential Operators on A+.- § 4.2. The Radial Component of the Casimir Operator.- § 4.3. Construction of the Eigenfunctions on G+.- § 4.4. The Harish-Chandra Series for ?? and the c-Function.- § 4.5. Estimates for the Harish-Chandra Series When ? Becomes Unbounded.- § 4.6. Estimates for the Elementary Spherical Functions. The Functions ? and ?.- § 4.7. The c-Function.- Noteson Chapter 4.- 5. Asymptotic Behaviour of Elementary Spherical Functions.- § 5.1. The Case When rk(G/K) =1.- § 5.2. The Basic Differential Equations Viewed as a Perturbation of a Linear System: The Regular Case.- § 5.3. Radial Components on M10? and M10+.- § 5.4. The Basic Differential Equations Viewed as a Perturbation of a Linear System: The General Case.- § 5.5. Spectral Theory of Representations of Polynomial Rings Associated to Finite Reflexion Groups.- § 5.6. The Initial Estimates.- § 5.7. Perturbations of Linear Systems (with A Priori Estimates).- § 5.8. Asymptotics of ?0(?:·) on M10+. The Function ?.- § 5.9. Asymptotics of ?(?:·).- § 5.10. Complements. Constant Term for Tempered ?-Finite Functions.- Notes on Chapter 5.- 6. The L2-Theory. The Harish-Chandra Transform on the Schwartz Space of G//K.- § 6.1. The Schwartz Spaces C(G) and C(G//K).- § 6.2. The Harish-Chandra Transform on C(G//K).- § 6.3. Wave Packets in C(G//K).- § 6.4. Statements of the Main Theorems.- § 6.5. The Method of Harish-Chandra.- § 6.6. The Method of Gangolli-Helgason-Rosenberg.- Notes on Chapter 6.- 7. Lp-Theory of Harish-Chandra Transform. Fourier Analysis on the Spaces Cp(G//K).- §7.1. Radial Components and Their Expansions.- § 7.2. The Differential Equations, Initial Estimates, and the Approximating Sequence.- § 7.3. Expressions for ?0 - ?n0, ?n0 - ?n-10, and Estimates for ?0 - exp(?0)?n0.- § 7.4. Further Study of the ?n0. The Matrices ?q.- § 7.5. The Functions ?q.- § 7.6. Asymptotic Expansions for ??.- § 7.7. The Tube Domains ??, *?? and the Function Spaces ?(??), ?(??).- § 7.8. The Spaces Cp(G//K).- § 7.9. Study of the Functions ?q.- § 7.10. Wave Packets and the Transform Theory for Cp(G//K).- Notes on Chapter 7.