  We study the role of long range dipolar interactions on antiferromagnetic
spin chains, from the classical $S\to \infty$ limit to the deep quantum case
$S=1/2$, including a transverse magnetic field. To this end, we combine
different techniques such as classical energy minima, classical Monte Carlo,
linear spin waves, bosonization and DMRG. We find a phase transition from the
already reported dipolar ferromagnetic region to an antiferromagnetic region
for high enough antiferromagnetic exchange. Thermal and quantum fluctuations
destabilize the classical order before reaching magnetic saturation in both
phases, and also close to zero field in the antiferromagnetic phase. In the
extreme quantum limit $S=1/2$, extensive DMRG computations show that the main
phases remain present with transition lines to saturation significatively
shifted to lower fields, in agreement with the bosonization analysis. The
overall picture keeps close analogy with the phase diagram of the anisotropic
XXZ spin chain in a transverse field.
