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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${}^{\ast }$(Sgr A${}^{\ast }$), connecting the size of the bright ring of emission to that of the underlying BH shadow and exploiting high-precision measurements of Sgr A${}^{\ast }$’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${}^{\ast }$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${}^{\ast }$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.