Two currently most competitive methods of measuring the Hubble constant based on observations of the cosmic microwave background radiation and type Ia supernovae yield strongly discrepant results. Since no unaccounted systematic effects have been found so far, there is now a growing conviction that the discrepancy -- dubbed the Hubble constant tension -- signifies an anomaly of the standard cosmological model. In my talk, I will review both observational and theoretical approaches to resolving the Hubble constant tension and understanding its origin. On the observational side, I will demonstrate the potential of several new methods to obtain alternative constraints on the Hubble constant, which will eventually confirm or raise doubts about the cosmological anomaly scenario.
In particular, I will show the first results from two novel techniques deriving the Hubble constant from dedicated observations of gravitational waves and extragalactic gamma rays. In the context of developing alternative methods to measure the Hubble constant, I will talk about ongoing searches of gravitationally lensed supernovae, their motivations and challenges. Finally, on the theoretical side I will make a critical review of possible extensions of the standard cosmological model proposed recently to alleviate the Hubble constant tension.