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We consider the dependence on parameters of the solutions of cohomologyequations over Anosov diffeomorphisms. We show that the solutions depend onparameters as smoothly as the data. As a consequence we prove optimalregularity results for the solutions of equations taking value indiffeomorphism groups. These results are motivated by applications to rigiditytheory, dynamical systems, and geometry. In particular, in the context ofdiffeomorphism groups we show: Let $f$ be a transitive Anosov diffeomorphism ofa compact manifold $M$. Suppose that $\eta \in C^{\reg}(M,\Diff^r(N))$ for acompact manifold $N$, $k,r \in \N$, $r \geq 1$, and $0 < \alpha \leq \Lip$. Weshow 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*} thenin fact $\varphi \in C^{\reg}(M,\Diff^r(N))$.

Smooth dependence on parameters of solution of cohomology equations over  Anosov systems and applications to cohomology equations on diffeomorphism  groups