Gravitational-wave (GW) cosmology provides a new way to measure the expansion history of the Universe and test General Relativity (GR) at cosmological scales in the tensor sector, based on the fact that GWs are direct distance tracers. Obtaining the redshift (whose knowledge is essential to test cosmology) is instead the challenge of GW cosmology. In absence of a direct electromagnetic counterpart to the GW event, the source goes under the name of ``dark siren'' and statistical techniques are used. This talk aims at giving an overview of the state-of-the-art for these techniques as well as discussing perspectives for the future. After introducing GW cosmology and statistical methods, I will present the latest measurements of the Hubble parameter and of the phenomenon of ``modified GW propagation'' (that takes place whenever GR is modified at cosmological scales), obtained from the latest Gravitational Wave Transient Catalog 3 with new, independent, open-source codes. In particular, the two techniques applied to real data so far consist in using the statistical correlation with galaxy catalogues and information from the mass distribution of Binary Black Holes. I will discuss methodological aspects, relevant sources of systematics, the interplay with population studies, current challenges and possible ways forward. I will finally present some not-yet-applied ideas for statistical dark siren techniques, in particular for third generation (3G) ground-based GW detectors.