Neutron stars offer us an excellent testbed to probe nuclear physics and strong-field gravity. Unlike the well-studied mass-radius relation for neutron stars that depends strongly on their internal structure, I first report unexpected universal relations that we found among the moment of inertia, tidal Love number and quadrupole moment ("I-Love-Q" relations) that are insensitive to the internal structure. Such universal relations help us break the degeneracy among neutron star parameters when probing fundamental physics with radio, X-ray or gravitational wave observations. I then explain similar universal relations among neutron star multipole moments, which suggest that such moments can be approximately described by the first three, namely the mass, spin and quadrupole moment. Such three-hair relations for neutron stars resemble the no-hair property of black holes. I finally describe a connection between the relations for "follicly challenged" neutron stars to those for bald black holes by studying slowly-rotating neutron star solutions with anisotropic pressure as a toy model. We found that the neutron star relations approach the black hole ones in a very non-trivial way as one increases the stellar compactness, which might be related to second order phase transitions in condensed matter physics.
