Gravitational wave (GW) observations are expected to serve as a powerful and independent probe of the expansion history of the universe. By providing direct and calibration-free measurements of luminosity distances through waveform analysis, GWs provide a fundamentally different and potentially more robust approach to measuring cosmic-scale distances compared to traditional electromagnetic (EM) observations, which is known as the standard siren method. In this review, we present an overview of recent developments in GW standard siren cosmology, including up-to-date H0 constraints: the re-analysis bright siren GW170817 \(H_{0}=78.4_{-12.0}^{+25.7}\text{km s}^{-1}\ \text{Mpc}^{-1}\) (employing the same methodology as the O4a dark and spectral siren studies), the most recent O4a dark-siren analysis \(H_{0}=81.6_{-15.9}^{+21.5}\text{km s}^{-1}\ \text{Mpc}^{-1}\) , and their combination \(H_{0}=76.6_{-9.5}^{+13.0}\text{km s}^{-1}\ \text{Mpc}^{-1}\) , and prospects for constraining cosmological parameters using future GW detections (H0 is expected to be constrained to the sub-percent level in a 10-year observation of the third-generation GW detectors). We first introduce standard sirens based on how redshift information is obtained and outline the Bayesian framework used in cosmological parameter estimation. We then review the measurements on the Hubble constant from the LIGO-Virgo-KAGRA network and present the potential role of future standard siren observations in cosmological parameter estimations. A central focus of this review is the unique ability of GW observations to break cosmological parameter degeneracies inherent in the EM observations. Since the cosmological parameter degeneracy directions of GW and EM observations are quite different (roughly orthogonal in some cases), their combination can significantly improve constraints on cosmological parameters. This complementarity is expected to become one of the most critical advantages for GW standard siren cosmology. We also briefly highlight the impact of systematic uncertainties, such as detector calibration, weak lensing, peculiar velocities, and host-galaxy catalog completeness, and corresponding potential mitigation strategies, which currently limit the constraint precision of cosmological parameters. Looking forward, we highlight the importance of combining GW standard sirens with other emerging late-universe cosmological probes such as fast radio bursts, 21 cm intensity mapping, and strong gravitational lensing to forge a precise cosmological probe for exploring the late universe. Finally, we introduce the challenges and the role of machine learning in searching for more signals, ensuring reliable parameter inferences, and accelerating the inference process for cosmological parameters.
