  We show that the Hitchin integrable system for a simple complex Lie group $G$
is dual to the Hitchin system for the Langlands dual group $\lan{G}$. In
particular, the general fiber of the connected component $\Higgs_0$ of the
Hitchin system for $G$ is an abelian variety which is dual to the corresponding
fiber of the connected component of the Hitchin system for $\lan{G}$. The
non-neutral connected components $\Higgs_{\alpha}$ form torsors over
$\Higgs_0$. We show that their duals are gerbes over $\Higgs_0$ which are
induced by the gerbe of $G$-Higgs bundles $\gHiggs$. More generally, we
establish a duality between the gerbe $\gHiggs$ of $G$-Higgs bundles and the
gerbe $\lan{\gHiggs}$ of $\lan{G}$-Higgs bundles, which incorporates all the
previous dualities. All these results extend immediately to an arbirtary
connected complex reductive group $\mathbb{G}$.
