  The $\Lambda$CDM model provides a good fit to a large span of cosmological
data but harbors areas of phenomenology. With the improvement of the number and
the accuracy of observations, discrepancies among key cosmological parameters
of the model have emerged. The most statistically significant tension is the
$4-6\sigma$ disagreement between predictions of the Hubble constant $H_0$ by
early time probes with $\Lambda$CDM model, and a number of late time,
model-independent determinations of $H_0$ from local measurements of distances
and redshifts. The high precision and consistency of the data at both ends
present strong challenges to the possible solution space and demand a
hypothesis with enough rigor to explain multiple observations--whether these
invoke new physics, unexpected large-scale structures or multiple, unrelated
errors. We present a thorough review of the problem, including a discussion of
recent Hubble constant estimates and a summary of the proposed theoretical
solutions. Some of the models presented are formally successful, improving the
fit to the data in light of their additional degrees of freedom, restoring
agreement within $1-2\sigma$ between {\it Planck} 2018, using CMB power spectra
data, BAO, Pantheon SN data, and R20, the latest SH0ES Team measurement of the
Hubble 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 additional
physics, pointing to the need for additional probes. While no specific proposal
makes 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 gravity
provide the best options until a better alternative comes along.[Abridged]
