  We study the moduli spaces of flat SL(r)- and PGL(r)-connections, or
equivalently, Higgs bundles, on an algebraic curve. These spaces are noncompact
Calabi-Yau orbifolds; we show that they can be regarded as mirror partners in
two different senses. First, they satisfy the requirements laid down by
Strominger-Yau-Zaslow (SYZ), in a suitably general sense involving a B-field or
flat unitary gerbe. To show this, we use their hyperkahler structures and
Hitchin's integrable systems. Second, their Hodge numbers, again in a suitably
general sense, are equal. These spaces provide significant evidence in support
of SYZ. Moreover, they throw a bridge from mirror symmetry to the duality
theory of Lie groups and, more broadly, to the geometric Langlands program.
