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.