The aim of the talk is to develop a Picard-Vessiot theory for differential equations in positive characteristic which is quite analogue to the characteristic zero theory. This is achieved by substituting ordinary differential equations by differential modules and equations for higher (iterative) derivations. With this modification it is possible to solve the inverse Galois problem in positive characteristic at least for connected linear groups and to prove an analogue of the Abhyankar conjecture. This finally explains some of the Ramis-Raynaud analogies. The lecture is based on common work with M. van der Put.
B.H. Matzat, M. v.d.Put: Iterative differential equations and the Abhyankar conjecture. Preprint 2001.