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We construct positive-genus analogues of Welschinger's invariants for manyreal symplectic manifolds, including the odd-dimensional projective spaces andthe renowned quintic threefold. In some cases, our invariants provide lowerbounds for counts of real positive-genus curves in real algebraic varieties.Our approach to the orientability problem is based entirely on the topology ofreal bundle pairs over symmetric surfaces; the previous attempts involveddirect computations for the determinant lines of Fredholm operators overbordered surfaces. We use the notion of real orientation introduced in thispaper to obtain isomorphisms of real bundle pairs over families of symmetricsurfaces and then apply the determinant functor to these isomorphisms. Thisallows us to endow the uncompactified moduli spaces of real maps from symmetricsurfaces of all topological types with natural orientations and to verify thatthey extend across the codimension-one boundaries of these spaces, thusimplementing a far-reaching proposal from C.-C. Liu's thesis for a fullyfledged real Gromov-Witten theory. The second and third parts of this workconcern applications: they describe important properties of our orientations onthe moduli spaces, establish some connections with real enumerative geometry,provide the relevant equivariant localization data for projective spaces, andobtain vanishing results in the spirit of Walcher's predictions.

Real Gromov-Witten Theory in All Genera and Real Enumerative Geometry: Construction