  The main purpose of the present paper is a study of orientations of the
moduli spaces of pseudo-holomorphic discs with boundary lying on a \emph{real}
Lagrangian submanifold, i.e., the fixed point set of an anti-symplectic
involutions $\tau$ on a symplectic manifold. We introduce the notion of
$\tau$-relatively spin structure for an anti-symplectic involution $\tau$, and
study how the orientations on the moduli space behave under the involution
$\tau$. We also apply this to the study of Lagrangian Floer theory of real
Lagrangian submanifolds. In particular, we study unobstructedness of the
$\tau$-fixed point set of symplectic manifolds and in particular prove its
unobstructedness in the case of Calabi-Yau manifolds. And we also do explicit
calculation of Floer cohomology of $\R P^{2n+1}$ over $\Lambda_{0,nov}^{\Z}$
which provides an example whose Floer cohomology is not isomorphic to its
classical cohomology. We study Floer cohomology of the diagonal of the square
of a symplectic manifold, which leads to a rigorous construction of the quantum
Massey product of symplectic manifold in complete generality.
