Summary of cosmology with gravitational waves from compact binary coalescences

GW170817 with its coincident optical counterpart has led to a first "standard siren" measurement of the Hubble constant independent of the cosmological distance ladder. The Schutz "statistical" method, which is expected to work in the absence of uniquely identified hosts, has also started bringing in its first estimates. In this work we report the current results of the gravitational-wave measurement of the Hubble constant and discuss the prospects with observations during the upcoming runs of the Advanced LIGO-Virgo detector network.