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This book is an introduction to the algebraic, algorithmic and analytic aspects of the Galois theory of homogeneous linear differential equations. Although the Galois theory has its origins in the 19th Century and was put on a firm footing by Kolchin in the middle of the 20th Century, it has experienced a burst of activity in the last 30 years. In this book we present many of the recent results and new approaches to this classical field. We have attempted to make this subject accessible to anyone with a background in algebra and analysis at the level of a first year graduate student. Our hope is that this book will prepare and entice the reader to delve further