In an accreting X-ray pulsar, a neutron star accretes matter from a companion star through an accretion disk. The magnetic field of the rotating neutron star disrupts the inner edge of the disk, funnelling the gas to flow onto the poles on its surface. Hercules X-1 is a prototypical persistent X-ray pulsar about 7 kpc from Earth. Its emission varies on three distinct timescales: the neutron star rotates every 1.2 s, it is eclipsed by its companion each 1.7 d, and the system exhibits a superorbital period of 35 d, which has remained stable since its discovery. Several lines of evidence point to the source of this variation as the precession of the accretion disk or that of the neutron star. Despite the many hints over the past 50 yr, the precession of the neutron star itself has yet not been confirmed or refuted. X-ray polarization measurements (probing the spin geometry of Her X-1) with the Imaging X-ray Polarimetry Explorer suggest that free precession of the neutron star crust sets the 35 d period; this has the important implication that its crust is somewhat asymmetric by a few parts per ten million.