I: Moduli Spaces.- § 1 Introduction.- § 2 Generalities about moduli-Spaces.- § 3 Examples.- § 4 Metrics with logarithmic singularities.- § 5 The minimal compact if ication of Ag/?.- § 6 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.- § 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 bundies.- § 2 Arakelov-divisors and intersection theory.- § 3 Volume forms on IRr(X, ?).- § 4 Riemann-Roch.- § 5 The Hodge index theorem.