  We construct positive-genus analogues of Welschinger's invariants for many
real symplectic manifolds, including the odd-dimensional projective spaces and
the renowned quintic threefold. In some cases, our invariants provide lower
bounds for counts of real positive-genus curves in real algebraic varieties.
Our approach to the orientability problem is based entirely on the topology of
real bundle pairs over symmetric surfaces; the previous attempts involved
direct computations for the determinant lines of Fredholm operators over
bordered surfaces. We use the notion of real orientation introduced in this
paper to obtain isomorphisms of real bundle pairs over families of symmetric
surfaces and then apply the determinant functor to these isomorphisms. This
allows us to endow the uncompactified moduli spaces of real maps from symmetric
surfaces of all topological types with natural orientations and to verify that
they extend across the codimension-one boundaries of these spaces, thus
implementing a far-reaching proposal from C.-C. Liu's thesis for a fully
fledged real Gromov-Witten theory. The second and third parts of this work
concern applications: they describe important properties of our orientations on
the moduli spaces, establish some connections with real enumerative geometry,
provide the relevant equivariant localization data for projective spaces, and
obtain vanishing results in the spirit of Walcher's predictions.
