Abstract: Information about the history of cosmic expansion is encoded in the distanceredshift relationship of extragalactic objects. Gravitational wave detection allows observers to directly measure the luminosity distance of gravitational wave sources, while traditional astronomical observations using electromagnetic waves as messengers can measure redshift information of the gravitational wave sources; the two methods combined can be used to probe the history of the expanding universe. In 2017, LIGO and Virgo achieved the first measurement of the cosmic expansion rate using gravitational waves through multi-messenger observations of the binary neutron star merger event GW170817. This method, independent of traditional electromagnetic distance-measurement techniques, presents a promising avenue to address current cosmological tensions. This article introduces the fundamental principles of using gravitational waves to investigate cosmic expansion, reviews current observational results and future prospects, and discusses outstanding challenges in this emerging field.