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The 13 chapters of this book centre around the proof of Theorem 1 of Faltings' paper "Diophantine approximation onabelian varieties", Ann. Math.133 (1991) and together givean approach to the proof that is accessible to Ph.D-level students in number theory and algebraic geometry. Each chapter is based on an instructional lecture given by its author ata special conference for graduate students, on thetopic of Faltings' paper.