I. Theta Functions from an Analytic Viewpoint -- ยง 1. Preliminaries -- ยง 2. Plancherel Theorem for Rn -- ยง 3. The Group A(X) -- ยง4. The Irreducibility of U -- ยง 5. Induced Representations -- ยง 6. The Group Sp(X) -- ยง7. The Group B(X) -- ยง 8. Fock Representation -- ยง9. The Set G(X) -- ยง 10. The Discrete Subgroup ?L -- II. Theta Functions from a Geometric Viewpoint -- ยง 1. Hodge Decomposition Theorem for a Torus -- ยง 2. Theta Function of a Positive Divisor -- ยง 3. The Automorphy Factor u?(z) -- ยง4. The Vector Space L(Q, l, ?) -- ยง 5. A Change of the Canonical Base -- III Graded Rings of Theta Functions -- ยง 1. Graded Rings -- ยง 2. Algebraic and Integral Dependence -- ยง 3. Weierstrass Preparation Theorem -- ยง 4. Geometric Lemmas -- ยง 5. Automorphic Forms and Projective Embeddings -- ยง 6. Polarized Abelian Varieties -- ยง 7. Projective Embeddings -- ยง8. The Field of Abelian Functions -- IV. Equations Defining Abelian Varieties -- ยง 1. Theta Relations (Classical Forms) -- ยง 2. A New Formalism -- ยง 3. Theta Relations (Under the New Formalism) -- ยง4. The Ideal of Relations -- ยง 5. Quadratic Equations Defining Abelian Varieties -- V. Graded Rings of Theta Constants -- ยง 1. Theta Constants -- ยง2. Some Properties of ?(?)2 -- ยง 3. Holomorphic Mappings by Theta Constants -- ยง 4. The Classical Reduction Theory -- ยง 5. Modular Forms -- ยง 6. The Group of Characteristics -- ยง 7. Modular Varieties -- Sources -- Further References and Comments -- Index of Definitions