The first direct detections of gravitational waves by the Laser Interferometer Gravitational-wave Observatory initiated the era of gravitational wave astronomy. Gravitational waves serve as a new and independent probe of the Universe. The combination of gravitational waves with information from other messengers, such as electromagnetic emission from the binary neutron star inspiral, led to a more complete and accurate understanding of cosmology and astrophysics. ,My thesis is focused on gravitational wave multi-messenger astronomy. The most promising sources for current gravitational wave detectors are compact binary mergers, including the mergers of stellar mass binary black holes, binary neutron stars, and neutron star-black hole system. I investigated the detection rate of binary neutron star and neutron star-black hole mergers from observations of their potential electromagnetic emission. To facilitate the search for the electromagnetic counterparts and the host galaxies of compact binaries, I developed a rapid algorithm that reconstructs the sky direction and luminosity distance of binary mergers from their gravitational wave signals, and predicted the existence of well-localized events. In addition, I carried out a thorough study of how gravitational-wave observational selection effects influence electromagnetic follow-up.,In summary, I explored how to measure astrophysical and cosmological parameters with gravitational wave detections, and facilitated gravitational wave-electromagnetic follow-up through various approaches, paving the way for the future of gravitational wave astrophysics and cosmology.