| Algebraic Theory | p. 1 |
| Picard-Vessiot rings | p. 4 |
| Existence and uniqueness of Picard-Vessiot rings | p. 7 |
| The Galois group | p. 8 |
| Galois correspondence for difference equations | p. 16 |
| Difference modules and fibre functors | p. 23 |
| Algorithms for difference equations | p. 28 |
| Difference equations of order one | p. 28 |
| Difference equations in diagonal form | p. 31 |
| Difference equations of order two | p. 33 |
| The inverse problem for difference equations | p. 35 |
| The ring S of sequences | p. 45 |
| An excursion in positive characteristic | p. 52 |
| Generalities | p. 52 |
| Modules over K[T. T superscript -1] | p. 56 |
| Difference Galois groups | p. 57 |
| Comparing characteristic 0 and p | p. 58 |
| Difference modules over P | p. 60 |
| Classification of difference modules over P | p. 60 |
| The universal Picard-Vessiot ring of P | p. 63 |
| Fields of constants which are not algebraically closed | p. 65 |
| Automorphisms of the universal Picard-Vessiot ring of P | p. 65 |
| Difference equations over C((z[superscript -1])) and the formal Galois group | p. 66 |
| Analytic Theory | p. 68 |
| Classification and canonical forms | p. 71 |
| A classification of singularities | p. 71 |
| Canonical forms | p. 75 |
| Semi-regular difference equations | p. 77 |
| Introduction | p. 77 |
| Some easy asymptotics | p. 78 |
| The connection matrix of a semi-regular equation | p. 80 |
| The theorem of Malgrange and Sibuya | p. 84 |
| Regular difference equations | p. 86 |
| Inverse problems for semi-regular equations | p. 88 |
| Mild difference equations | p. 95 |
| Asymptotics for mild equations | p. 95 |
| Connection matrices of mild equations | p. 96 |
| Tame differential modules | p. 105 |
| Inverse problems for mild equations | p. 106 |
| Examples of equations and Galois groups | p. 111 |
| Calculating connection matrices | p. 111 |
| Classification of order one equations | p. 116 |
| More on difference Galois groups | p. 119 |
| Mild difference and differential equations | p. 122 |
| Very mild difference modules and multisummability | p. 124 |
| Very mild differential modules | p. 125 |
| Wild difference equations | p. 127 |
| Introduction | p. 127 |
| Multisummability of formal solutions | p. 128 |
| The Quadrant Theorem | p. 129 |
| On the Gamma function | p. 130 |
| An example | p. 131 |
| Solutions on a right half plane | p. 133 |
| Solutions on an upper half plane | p. 137 |
| Analytic equivalence classes of difference equations | p. 140 |
| An example | p. 144 |
| q-difference equations | p. 149 |
| Formal aspects | p. 149 |
| Analytic properties | p. 153 |
| Regular singular equations over k[subscript 0] | p. 154 |
| Equations over C(z) | p. 156 |
| Construction of the connection map | p. 157 |
| Meromorphic vector bundles | p. 157 |
| The connection map of a regular equation | p. 159 |
| The connection map of a regular singular equation | p. 162 |
| Inverse problems | p. 166 |
| Bibliography | p. 175 |
| Index | p. 179 |
| Notations | p. 180 |
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