We consider the problem of expressing the regularity of different objects which appear in Anosov systems in terms of Sobolev spaces. We show in examples that one cannot expect any improvements in the regularity of Anosov splitting from the classical results that give regularity in W^{r_0,\infty}. We also develop a Sobolev regularity theory for solutions of cohomology equations, both commutative and non-commutative, and the bootstrap of regularity arguments from Sobolev to other classes for a wide variety of functional equations involving Anosov systems.