  We report on recent results obtained in the post-Newtonian framework for the
modelling of the gravitational waves emitted by binary systems of spinning
compact objects (black holes and/or neutron stars). These new results are
obtained at the spin-orbit (linear-in-spin) level and solving Einstein's field
equations iteratively in harmonic coordinates as well as the multipolar
post-Newtonian formalism. The dynamics of the binary was tackled at the
next-to-next-to-leading order, corresponding to the 3.5 post-Newtonian (PN)
order for maximally spinning objects, and the result is found to be consistent
with a previously obtained reduced Hamiltonian in the ADM approach. The
corresponding contribution to the energy flux emitted by the binary was
obtained at the 3.5PN order, as well as the next-to-leading 4PN tail
contribution to this flux, an imprint of the non-linearity in the propagation
of the wave. These new terms can be used to build more accurate PN templates
for the next generation of gravitational wave detectors. We give an
illustrative estimate of the quantitative relevance of the new terms in the
orbital phasing of the binary.
