  We consider the dependence on parameters of the solutions of cohomology
equations over Anosov diffeomorphisms. We show that the solutions depend on
parameters as smoothly as the data. As a consequence we prove optimal
regularity results for the solutions of equations taking value in
diffeomorphism groups. These results are motivated by applications to rigidity
theory, dynamical systems, and geometry. In particular, in the context of
diffeomorphism groups we show: Let $f$ be a transitive Anosov diffeomorphism of
a compact manifold $M$. Suppose that $\eta \in C^{\reg}(M,\Diff^r(N))$ for a
compact manifold $N$, $k,r \in \N$, $r \geq 1$, and $0 < \alpha \leq \Lip$. We
show that if there exists a $\varphi\in C^{\reg}(M,\Diff^1(N))$ solving
\begin{equation*} \varphi_{f(x)} = \eta_x \circ \varphi_x \end{equation*} then
in fact $\varphi \in C^{\reg}(M,\Diff^r(N))$.
