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The $\Lambda$CDM model provides a good fit to a large span of cosmologicaldata but harbors areas of phenomenology. With the improvement of the number andthe accuracy of observations, discrepancies among key cosmological parametersof the model have emerged. The most statistically significant tension is the$4-6\sigma$ disagreement between predictions of the Hubble constant $H_0$ byearly time probes with $\Lambda$CDM model, and a number of late time,model-independent determinations of $H_0$ from local measurements of distancesand redshifts. The high precision and consistency of the data at both endspresent strong challenges to the possible solution space and demand ahypothesis with enough rigor to explain multiple observations--whether theseinvoke new physics, unexpected large-scale structures or multiple, unrelatederrors. We present a thorough review of the problem, including a discussion ofrecent Hubble constant estimates and a summary of the proposed theoreticalsolutions. Some of the models presented are formally successful, improving thefit to the data in light of their additional degrees of freedom, restoringagreement within $1-2\sigma$ between {\it Planck} 2018, using CMB power spectradata, BAO, Pantheon SN data, and R20, the latest SH0ES Team measurement of theHubble constant ($H_0 = 73.2 \pm 1.3{\rm\,km\,s^{-1}\,Mpc^{-1}}$ at 68\%confidence level). Reduced tension might not simply come from a change in $H_0$but also from an increase in its uncertainty due to degeneracy with additionalphysics, pointing to the need for additional probes. While no specific proposalmakes a strong case for being highly likely or far better than all others,solutions involving early or dynamical dark energy, neutrino interactions,interacting cosmologies, primordial magnetic fields, and modified gravityprovide the best options until a better alternative comes along.[Abridged]

In the Realm of the Hubble tension $-$ a Review of Solutions