1 Introduction to Locally Convex Topological Vector Spaces and Dual Pairs.- §1. Locally Convex Vector Spaces.- §2. Hahn-Banach Theorems.- §3. Dual Pairs.- Notes and Remarks.- 2 Radon Measures and Integral Representations.- §1. Introduction to Radon Measures on Hausdorff Spaces.- §2. The Riesz Representation Theorem.- §3. Weak Convergence of Finite Radon Measures.- §4. Vague Convergence of Radon Measures on Locally Compact Spaces.- §5. Introduction to the Theory of Integral Representations.- Notes and Remarks.- 3 General Results on Positive and Negative Definite Matrices and Kernels.- §1. Definitions and Some Simple Properties of Positive and Negative Definite Kernels.- §2. Relations Between Positive and Negative Definite Kernels.- §3. Hubert Space Representation of Positive and Negative Definite Kernels.- Notes and Remarks.- 4 Main Results on Positive and Negative Definite Functions on Semigroups.- §1. Definitions and Simple Properties 86 §2. Exponentially Bounded Positive Definite Functions on Abelian Semigroups.- §3. Negative Definite Functions on Abelian Semigroups.- §4. Examples of Positive and Negative Definite Functions.- §5. T-Positive Functions.- §6. Completely Monotone and Alternating Functions.- Notes and Remarks.- 5 Schoenberg-Type Results for Positive and Negative Definite Functions.- §1. Schoenberg Triples.- §2. Norm Dependent Positive Definite Functions on Banach Spaces.- §3. Functions Operating on Positive Definite Matrices.- §4. Schoenberg's Theorem for the Complex Hilbert Sphere.- §5. The Real Infinite Dimensional Hyperbolic Space.- Notes and Remarks.- 6 Positive Definite Functions and Moment Functions.- §1. Moment Functions.- §2. The One-Dimensional Moment Problem.- §3. The Multi-Dimensional Moment Problem.- §4. TheTwo-Sided Moment Problem.- §5. Perfect Semigroups.- Notes and Remarks.- 7 Hoeffding's Inequality and Multivariate Majorization.- §1. The Discrete Case.- §2. Extension to Nondiscrete Semigroups.- §3. Completely Negative Definite Functions and Schur-Monotonicity.- Notes and Remarks.- 8 Positive and Negative Definite Functions on Abelian Semigroups Without Zero.- §1. Quasibounded Positive and Negative Definite Functions.- §2. Completely Monotone and Completely Alternating Functions.- Notes and Remarks.- References.- List of Symbols.