This collection presents the proceedings of the conference on arithmetic and differential Galois groups. On almost 400 pages there are 15 contributions delivered at the conference showing vivid interactions between two highly developed theories: the Galois theory of algebraic functions and the theory of differential operators. The range of topics covered contains expositions on many of the problems in this field. The reader will find here a wealth of interesting material on: arithmetic differential algebra and its applications to the invariant theory of correspondences; the geometry behind the sixth Painlevé differential equation and the construction of some its algebraic solutions; the Abhyankar conjecture, explicit computations and the realisation of Galois groups; and how construction problems of algebraic varieties with various topological behaviour motivate the search for methods of doing mathematics in a machine checked way. The collection of papers will certainly help anybody interested in the subject get oriented, and will provide them with stimulating impetus for further reading or motivation for future research.