This book consists of the notes from the seminar Bonn/ Wuppertal 1983/ 84 on Arithmetic Geometry. It contains a proof for the Mordell conjecture and may be useful as an introduction to Arakelov's point of view in diophantine geometry. The third edition includes an appendix in which a detailed survey on the spectacular recent developments in arithmetic algebraic geometry is given. These beautiful new results have their roots in the material covered by this book.

I: Moduli Spaces.- § 1 Introduction.- § 2 Generalities about moduli spaces.- § 3 Examples.- § 4 Metrics with logarithmic singularities.- § 5 The minimal compactification of Ag/?.- § 8 The toroidal compactification.- II: Heights.- § 1 The definition.- § 2 Néron-Tate heights.- § 3 Heights on the moduli space.- § 4 Applications.- III: Some Facts from the Theory of Group Schemes.- § 0 Introduction.- § 1 Generalities on group schemes.- § 2 Finite group schemes.- § 3 p-divisible groups.- § 4 A theorem of Raynaud.- § 5 A theorem of Tate.- IV: Tate's Conjecture on the Endomorphisms of Abelian Varieties.- § 1 Statements.- § 2 Reductions.- § 3 Heights.- § 4 Variants.- V: The Finiteness Theorems of Faltings.- § 1 Introduction.- § 2 The finiteness theorem for isogeny classes.- § 3 The finiteness theorem for isomorphism classes.- § 4 Proof of Mordell's conjecture.- § 5 Siegel's Theorem on integer points.- VI: Complements to Mordell.- § 1 Introduction.- § 2 Preliminaries.- § 3 The Tate conjecture.-§ 4 The Shafarevich conjecture.- § 5 Endomorphisms.- § 6 Effectivity.- VII: Intersection Theory on Arithmetic Surfaces.- § 0 Introduction.- § 1 Hermitian line bundles.- § 2 Arakelov divisors and intersection theory.- § 3 Volume forms on IR?(X, ?).- § 4 Riemann Roch.- § 5 The Hodge index theorem.- Appendix: New Developments in Diophantine and Arithmetic Algebraic Geometry (Gisbert Wüstholz).- § 2 The transcendental approach.- § 3 Vojta's approach.- § 4 Arithmetic Riemann-Roch Theorem.- § 5 Applications in Arithmetic.- § 6 Small sections.- § 7 Vojta's proof in the number field case.- § 8 Lang's conjecture.- § 9 Proof of Faltings' theorem.- § 10 An elementary proof of Mordell's conjecture.- § 11 ?-adic representations attached to abelian varieties.