  Horizon-scale images of black holes (BHs) and their shadows have opened an
unprecedented window onto tests of gravity and fundamental physics in the
strong-field regime. We consider a wide range of well-motivated deviations from
classical General Relativity (GR) BH solutions, and constrain them using the
Event Horizon Telescope (EHT) observations of Sagittarius A$^*$ (Sgr A$^*$),
connecting the size of the bright ring of emission to that of the underlying BH
shadow and exploiting high-precision measurements of Sgr A$^*$'s
mass-to-distance ratio. The scenarios we consider, and whose fundamental
parameters we constrain, include various regular BHs, string-inspired
space-times, violations of the no-hair theorem driven by additional fields,
alternative theories of gravity, novel fundamental physics frameworks, and BH
mimickers including well-motivated wormhole and naked singularity space-times.
We demonstrate that the EHT image of Sgr A$^*$ places particularly stringent
constraints on models predicting a shadow size larger than that of a
Schwarzschild BH of a given mass, with the resulting limits in some cases
surpassing cosmological ones. Our results are among the first tests of
fundamental physics from the shadow of Sgr A$^*$ and, while the latter appears
to be in excellent agreement with the predictions of GR, we have shown that a
number of well motivated alternative scenarios, including BH mimickers, are far
from being ruled out at present.
