  The present note overviews our recent construction of real Gromov-Witten
theory in arbitrary genera for many real symplectic manifolds, including the
odd-dimensional projective spaces and the renowned quintic threefold, its
properties, and its connections with real enumerative geometry. Our
construction introduces the principle of orienting the determinant of a
differential operator relative to a suitable base operator and a real setting
analogue of the (relative) spin structure of open Gromov-Witten theory.
Orienting the relative determinant, which in the now-standard cases is
canonically equivalent to orienting the usual determinant, is naturally related
to the topology of vector bundles in the relevant category. This principle and
its applications allow 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.
