Gravitational-wave (GW) astronomy has opened a whole new window of looking at the Universe, and has revolutionized our understanding of compact binary sources that these waves arise from. In this seminar, I shall explore a few possibilities of probing gravity, cosmology, and astrophysics using observations of GWs from current and future detectors. This will include 1) developing a method to place constraints on the time variation of the gravitational constant G using the detected LIGO-Virgo binary neutron star (BNS) mergers, 2) asking whether a BNS merger near a supermassive black hole (SMBH).could enable a detection of the elusive post-merger signal, 3) investigating how well we can measure the imprint of centre-of-mass acceleration on a GW signal, and 4) develop a method to measure the large-scale two-point correlation function using GW observations in proposed next-generation detectors, accounting for localization uncertainties and realistic merger rates.