Difference algebraic groups are subgroups of the general linear group defined by a system of algebraic difference equations in the matrix entries. These groups have a rich structure theory, to some extent analogous to the theory of linear algebraic groups.

Difference algebraic groups have applications in number theory and occur as Galois groups of functional equations depending on a parameter. I will explain some structure results and decomposition theorems for these groups and I will show how these results are applied in the study of the relations among the solutions of a linear differential equation and their transforms under various operations like scaling or shifting.