  The Torelli group of a compact non-orientable Klein surface is the subgroup
of the modular group consisting of the mapping classes that act trivially on
the first homology group of the surface. We prove that if a surface has genus
at least $3$, then the Torelli group acts fixed points free on the
Teichm\"{u}ller space of the surface. That gives an embedding of the Torelli
space of a Klein surface in the Torelli space of its complex double. We also
construct real tori associated to Klein surfaces, which we call the Jacobian of
the surface. We prove that this Jacobian is isomorphic to a component of the
real part of the Jacobian of the complex double.
