  The recent direct observation of gravitational wave event $GW170817$ and its
$GRB170817A$ signal has opened up a new window to study neutron stars and
heralds a new era of Astronomy referred to as the Multimessenger Astronomy.
Both gravitational and electromagnetic waves from a single astrophysical source
have been detected for the first time. This combined detection offers an
unprecedented opportunity to place constraints on the neutron star matter
equation of state. The existence of a possible hadron-quark phase transition in
the central regions of neutron stars is associated with the appearance of
g-modes, which are extremely important as they could signal the presence of a
pure quark matter core in the centers of neutron stars. Observations of g-modes
with frequencies between 1 kHz and 1.5 kHz could be interpreted as evidence of
a sharp hadron-quark phase transition in the cores of neutron stars. In this
article, we shall review the description of the dense matter composing neutron
stars, the determination of the equation of state of such matter, and the
constraints imposed by astrophysical observations of these fascinating compact
objects.
