I. The Galton-Watson Process.- A. Preliminaries.- B. A First Look at Limit Theorems.- C. Finer Limit Theorems.- D. Further Ramifications.- Complements and Problems I.- II. Potential Theory.- 1. Introduction.- 2. Stationary Measures: Existence, Uniqueness, and Representation.- 3. The Local Limit Theorem for the Critical Case.- 4. The Local Limit Theorem for the Supercritical Case.- 5. Further Properties of W; A Sharp Global Limit Law; Positivity of the Density.- 6. Asymptotic Properties of Stationary Measures.- 7. Green Function Behavior.- 8. Harmonic Functions.- 9. The Space-Time Boundary.- Complements and Problems II.- III. One Dimensional Continuous Time Markov Branching Processes.- 1. Definition.- 2. Construction.- 3. Generating Functions.- 4. Extinction Probability and Moments.- 5. Examples: Binary Fission, Birth and Death Process.- 6. The Embedded Galton-Watson Process and Applications to Moments.- 7. Limit Theorems.- 8. More on Generating Functions.- 9. Split Times.- 10. Second Order Properties.- 11. Constructions Related to Poisson Processes.- 12. The Embeddability Problem.- Complements and Problems III.- IV. Age-Dependent Processes.- 1. Introduction.- 2. Existence and Uniqueness.- 3. Comparison with Galton-Watson Process; Embedded Generation Process; Extinction Probability.- 4. Renewal Theory.- 5. Moments.- 6. Asymptotic Behavior of F(s, t) in the Critical Case.- 7. Asymptotic Behavior of F(s, t) when m?1: The Malthusian Case.- 8. Asymptotic Behavior of F(s, t) when m?1: Sub-Exponential Case.- 9. The Exponential Limit Law in the Critical Case.- 10. The Limit Law for the Subcritical Age-Dependent Process.- 11. Limit Theorems for the Supercritical Case.- Complements and Problems IV.- V. Multi-Type Branching Processes.- 1. Introduction and Definitions.- 2.Moments and the Frobenius Theorem.- 3. Extinction Probability and Transience.- 4. Limit Theorems for the Subcritical Case.- 5. Limit Theorems for the Critical Case.- 6. The Supercritical Case and Geometric Growth.- 7. The Continuous Time, Multitype Markov Case.- 8. Linear Functionals of Supercritical Processes.- 9. Embedding of Urn Schemes into Continuous Time Markov Branching Processes.- 10. The Multitype Age-Dependent Process.- Complements and Problems V.- VI. Special Processes.- 1. A One Dimensional Branching Random Walk.- 2. Cascades; Distributions of Generations.- 3. Branching Diffusions.- 4. Martingale Methods.- 5. Branching Processes with Random Environments.- 6. Continuous State Branching Processes.- 7. Immigration.- 8. Instability.- Complements and Problems VI.- List of Symbols.- Author Index.