1. Valuation Rings. - 1. 1. Valuation Rings and Valuations of Fields. - 1. 2. Valuation Rings in Algebraic Extensions. - 1. 3. Henselian Valuation Rings. - 1. 4. Alebgraic Extensions of Valued Fields. - 1. 5. Immediate Extensions. - 1. 6. Density. - 1. 7. Constructions. - 2. Multi-Valued Fields. - 2. 1. Prüfer Rings. - 2. 2. The Zariski Topology and the Restriction Mapping. - 2. 3. Affine Families of Valuation Rings. - 2. 4. Weakly Boolean Families and Boolean Families of Valuation Rings. - 2. 5. Near Boolean Families of Valuation Rings. - 2. 6. Independence. - 3. Local-Global Properties of Near Boolean Families. - 3. 1. Rational Points over Henselian Fields. - 3. 2. Arithmetic Local-Global Principle. - 3. 3. Equivalent Forms of Property LGA. - 3. 4. Preservation of Property LGA under Algebraic Separable Extensions. - 3. 5. Geometric Local-Global Principle. - 3. 6. Existence of Families with Property LGA. - 4. Model-Theoretic Properties of Multi-Valued Fields. - 4. 1. Embedding Theorems. - 4. 2. The Robinson Theorem. - 4. 3. Valued Fields. - 4. 4. Finitely Multi-Valued Fields. - 4. 5. Multi-Valued Fields with Boolean Families. - 4. 6. Multi-Valued Fields with Near Boolean Families. - Bibliographical and Historical Remarks. - References. Language: English