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GW190521, the most massive binary black hole merger confidently detected by the LIGO-Virgo- KAGRA Collaboration, is the first gravitational-wave observation of an intermediate-mass black hole. The signal was followed approximately 34 days later by flare ZTF19abanrhr, detected in AGN J124942.3 þ 344929 by the Zwicky Transient Facility at the 78% spatial contour for GW190521’s sky localization. Using the GWTC-2.1 data release, we find that the association between GW190521 and flare ZTF19abanrhr as its electromagnetic counterpart is preferred over a random coincidence of the two transients with a log Bayes’ factor of 8.6, corresponding to an odds ratio of ∼5400∶1 for equal prior odds and ∼400∶1 assuming an astrophysical prior odds of 1=13. Given the association, the multimessenger signal allows for an estimation of the Hubble constant, finding H0 ¼ 102þ27 −25 km s−1 Mpc−1 when solely analyzing GW190521 and 79.2þ17.6 −9.6 km s−1 Mpc−1 assuming prior information from the binary neutron star merger GW170817, both consistent with the existing literature. 

Standard sirens have been the central paradigm in gravitational-wave cosmology so far. From the gravitational wave signature of compact star binaries, it is possible to measure the luminosity distance of the source directly, and if additional information on the source redshift is provided, a measurement of the cosmological expansion can be performed. This review article discusses several methodologies that have been proposed to use gravitational waves for cosmological studies. Methods that use only gravitational-wave signals and methods that use gravitational waves in conjunction with additional observations such as electromagnetic counterparts and galaxy catalogs will be discussed. The review also discusses the most recent results on gravitational-wave cosmology, starting from the binary neutron star merger GW170817 and its electromagnetic counterpart and finishing with the population of binary black holes, observed with the third Gravitational-wave Transient Catalog GWTC–3.