  Neutron stars and quark stars are ideal laboratories to study fundamental
physics at supra nuclear densities and strong gravitational fields.
Astrophysical observables, however, depend strongly on the star's internal
structure, which is currently unknown due to uncertainties in the equation of
state. Universal relations, however, exist among certain stellar observables
that do not depend sensitively on the star's internal structure. One such set
of relations is between the star's moment of inertia ($I$), its tidal Love
number (Love) and its quadrupole moment ($Q$), the so-called I-Love-Q
relations. Similar relations hold among the star's multipole moments, which
resemble the well-known black hole no-hair theorems. Universal relations break
degeneracies among astrophysical observables, leading to a variety of
applications: (i) X-ray measurements of the nuclear matter equation of state,
(ii) gravitational wave measurements of the intrinsic spin of inspiraling
compact objects, and (iii) gravitational and astrophysical tests of General
Relativity that are independent of the equation of state. We here review how
the universal relations come about and all the applications that have been
devised to date.
