  Einstein's theory of general relativity predicts that the only stationary
configuration of an isolated black hole is the Kerr spacetime, which has a
unique multipolar structure and a spherical shape when non-spinning. This is in
striking contrast to the case of other self-gravitating objects, which instead
can in principle have arbitrary deformations even in the static case. Here we
develop a general perturbative framework to construct stationary stars with
small axisymmetric deformations, and study explicitly compact stars with an
intrinsic quadrupole moment. The latter can be sustained, for instance, by
crust stresses or strong magnetic fields. While our framework is general, we
focus on quadrupolar deformations of neutron stars induced by an anisotropic
crust, which continuously connect to spherical neutron stars in the isotropic
limit. Deformed neutron stars might provide a more accurate description for
stellar remnants formed in supernovae and in binary mergers, and can be used to
improve constraints on the neutron-star equation of state through
gravitational-wave detections and through the observation of low-mass X-ray
binaries. We argue that, if the (dimensionless) intrinsic quadrupole moment is
of a few percent or higher, the effect of the deformation is stronger than that
of tidal interactions in coalescing neutron-star binaries, and might also
significantly affect the electromagnetic signal from accreting neutron stars.
