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: Galois theory of algebraic functions and the theory of differential operators. The range of topics covers expositions on many facets of the field. The reader will find here a wealth of interesting material: on arithmetic differential algebra and its applications to the invariant theory of correspondences; on the geometry behind the sixth Panlevé differential equation and construction of some its algebraic solutions; and on the Abhyankar conjecture, explicit computations or the realisation of Galois groups. Also contained in this collection is 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 orient anybody interested in the subject and will provide them with a stimulating impetus for further reading or motivation for future research.