Algebraic groups and differential Galois theory Differential Galois theory has seen intense research activity during the last decades in several directions: elaboration of more general theories computational aspects model theoretic approaches applications to classical and quantum mechanics as well as to other mathematical areas such as number theory. This book intends to introduce the reader to this subject by presenting Picard Vessiot theory i.e. Galois theory of linear differential equations in a self contained way. The needed prerequisites from algebraic geometry and algebraic groups are contained in the first two parts of the book. The third part includes Picard Vessiot extensions the fundamental theorem of Picard Vessiot theory solvability by quadratures Fuchsian equations monodromy group and Kovacics algorithm. Over one hundred exercises will help to assimilate the concepts and to introduce the reader to some topics beyond the scope of this book. Publishers description Differential Galois theory has seen intense research activity during the last decades in several directions: elaboration of more general theories computational aspects model theoretic approaches applications to classical and quantum mechanics as well as to other mathematical areas such as number theory. This book intends to introduce the reader to this subject by presenting Picard Vessiot theory i.e. Galois theory of linear differential equations in a self contained way. The needed prerequisites from algebraic geometry and algebraic groups are contained in the first two parts of the book. The third part includes Picard Vessiot extensions the fundamental theorem of Picard Vessiot theory solvability by quadratures Fuchsian equations monodromy group and Kovacics algorithm. Over one hundred exercises will help to assimilate the concepts and to introduce the reader to some topics beyond the scope of this book.