  We define Hitchin's moduli space for a principal bundle $P$, whose structure
group is a compact semisimple Lie group $K$, over a compact non-orientable
Riemannian manifold $M$. We use the Donaldson-Corlette correspondence, which
identifies Hitchin's moduli space with the moduli space of flat
$K^\mathbb{C}$-connections, which remains valid when M is non-orientable. This
enables us to study Hitchin's moduli space both by gauge theoretical methods
and algebraically by using representation varieties. If the orientable double
cover $\tilde{M}$ of $M$ is a K\"ahler manifold with odd complex dimension and
if the K\"ahler form is odd under the non-trivial deck transformation on
$\tilde{M}$, Hitchin's moduli space of the pull-back bundle $\tilde{P}$ over
$\tilde{M}$ has a hyper-K\"ahler structure and admits an involution induced by
the deck transformation. The fixed-point set is symplectic or Lagrangian with
respect to various symplectic structures on Hitchin's moduli space over
$\tilde{M}$. We show that there is a local diffeomorphism from Hitchin's moduli
space over (the nonorientable manifold) $M$ to the fixed point set of the
Hitchin's moduli space over (its orientable double cover) $\tilde{M}$. We
compare the gauge theoretical constructions with the algebraic approach using
representation varieties.
