The persistent discrepancy in the value of the Hubble constant (H₀) is one of the most significant current challenges to the standard ΛCDM cosmological model. While cosmic microwave background (CMB) measurements infer one value for H₀, local determinations based on the distance ladder yield a significantly different figure. This review focuses on theoretical responses seeking to resolve this inconsistency, analyzing how modifications can affect everything from gravitational field equations to the pre-recombination sound horizon and the late-time distance-redshift relation.

The study primarily examines modified gravity theories, as well as mechanisms operating in early or late stages of the universe. These mechanisms could alter the standard acoustic ruler, distance measurements, the growth of large-scale structures, or the gravitational response. Although some current proposals manage to reduce the nominal tension in H₀, they often do so at the expense of introducing correlated shifts in other cosmological observables, such as CMB spectra, standard ruler distances, gravitational lensing, or calibration information.

To advance in resolving the Hubble tension, the need to develop approaches that integrate data from multiple cosmological probes under the same set of model assumptions is emphasized. This involves creating unified likelihood functions and multiprobe tests that link all key observables. Only then can it be determined whether any of the proposed theoretical explanations can resolve the H₀ discrepancy without generating new inconsistencies in other areas of cosmology.