Arakelov introduced a component at infinity in arithmetic considerations, thus giving rise to global theorems similar to those of the theory of surfaces, but in an arithmetic context over the ring of integers of a number field. The book gives an introduction to this theory, including the analogues of the Hodge Index Theorem, the Arakelov adjunction formula, and the Faltings Riemann-Roch theorem. The book is intended for second year graduate students and researchers in the field who want a systematic introduction to the subject. The residue theorem, which forms the basis for the adjunction formula, is proved by a direct method due to Kunz and Waldi. The Faltings Riemann-Roch theorem is proved without assumptions of semistability. An effort has been made to include all necessary details, and as complete references as possible, especially to needed facts of analysis for Green's functions and the Faltings metrics.

I Metrics and Chern Forms.- §1. Néron Functions and Divisors.- §2. Metrics on Line Sheaves.- §3. The Chern Form of a Metric.- §4. Chern Forms in the Case of Riemann Surfaces.- II Green's Functions on Rlemann Surface.- §1. Green's Functions.- §2. The Canonical Green's Function.- §3. Some Formulas About the Green's Function.- §4. Coleman's Proof for the Existence of Green's Function.- §5. The Green's Function on Elliptic Curves.- III Intersection on an Arithmetic Surface.- §1. The Chow Groups.- §2. Intersections.- §3. Fibral Intersections.- §4. Morphisms and Base Change.- §5. Néron Symbols.- IV Hodge Index Theorem and the Adjunction Formula.- §1. Arakelov Divisors and Intersections.- §2. The Hodge Index Theorem.- §3. Metrized Line Sheaves and Intersections.- §4. The Canonical Sheaf and the Residue Theorem.- §5. Metrizations and Arakelov's Adjunction Formula.- V The Faltings Reimann-Roch Theorem.- §1. Riemann-Roch on an Arithmetic Curve.- §2. Volume Exact Sequences.- §3. Faltings Riemann-Roch.- §4. An Application of Riemann-Roch.- §5. Semistability.- §6. Positivity of the Canonical Sheaf.- VI Faltings Volumes on Cohomology.- §1. Determinants.- §2. Determinant of Cohomology.- §3. Existence of the Faltings Volumes.- §4. Estimates for the Faltings Volumes.- §5. A Lower Bound for Green's Functions.- Appendix by Paul Vojta Diophantine Inequalities and Arakelov Theory.- §1. General Introductory Notions.- §2. Theorems over Function Fields.- §3. Conjectures over Number Fields.- §4. Another Height Inequality.- §5. Applications.- References.- Frequently Used Symbols.