Classical Galois theory of algebraic equations has been extended to a Galois theory of linear differential equations [vS03] and more recently to the Galois theory of various kind of linear difference equations [vS97, DV21]. 
Apart from an intrinsic interest, the Galois theory of linear difference equations has been proven to have surprising applications in many different domains. First, I will start giving an introduction to the Galois theory of linear differential equations. Then I will continue presenting the Galois theory of linear functional equations and some of its applications, that have been the object of recent publications. The interactions between combinatorics, probability and Galois theory of functional equations is very promising: Many problems are still open and for this reason I think that it is a good subject for a post-graduate class.