This is an introduction to some geometrie aspects of G-function theory. Most of the results presented here appear in print for the flrst time; hence this text is something intermediate between a standard monograph and a research artic1e; it is not a complete survey of the topic. Except for geometrie chapters (I.3.3, II, IX, X), I have tried to keep it reasonably self­ contained; for instance, the second part may be used as an introduction to p-adic analysis, starting from a few basic facts wh ich are recalled in IV.l.l. I have inc1uded about forty exercises, most of them giving some complements to the main text. Acknowledgements This book was written during a stay at the Max-Planck-Institut in Bonn. I should like here to express my special gratitude to this institute and its director, F. Hirzebruch, for their generous hospitality. G. Wüstholz has suggested the whole project and made its realization possible, and this book would not exist without his help; I thank him heartily. I also thank D. Bertrand, E. Bombieri, K. Diederich, and S. Lang for their encouragements, and D. Bertrand, G. Christo I and H Esnault for stimulating conversations and their help in removing some inaccuracies after a careful reading of parts of the text (any remaining error is however my sole responsibility).

One: What are G-functions?.- I: G-functions.- II: Geometric differential equations.- Two: G-functions and differential equations.- III: Fuchsian differential systems: formal theory.- IV: Fuchsian differential systems: arithmetic theory.- V: Local methods.- VI: Global methods.- Three: Diophantine questions.- VII: Independence of values of G-functions.- VIII: A criterium of rationality.- Four: G-functions in arithmetic algebraic geometry.- IX: Towards Grothendieck's conjecture on periods of algebraic manifolds.- X: Endomorphisms in the fibers of an Abelian pencil.- Glossary of notations.