Neutron stars are a unique class of astronomical objects that are born after  stellar death of massive stars. They are the densest object found in nature,  with the strongest magnetic field, and a very strong gravity. However, their  internal composition has remained elusive even after half a century of their  discovery. The recent observation of binary neutron star merger event (GW170817)  has opened up a new window to our understanding of their composition by  constraining theoretically proposed equations of state. In this talk, I will describe  the difficulties faced with the previous effort with X-ray observations, and how  the recent gravitational wave observations played an important role in this front.  I will also present our ongoing endeavor of detecting continuous gravitational  sources from rapidly spinning neutron stars. I will then briefly mention what we  can learn about the nature of the neutron stars from this type of sources. Finally,  I will discuss how gravitational wave observations can potentially be used look  for the existence of a number of exotic compact objects, e.g., Boson stars, that  can potentially mimic mergers of binary black hole systems.