We perform a comprehensive study of Milky Way (MW) satellite galaxies to constrain the fundamental properties of dark matter (DM). This analysis fully incorporates inhomogeneities in the spatial distribution and detectability of MW satellites, and marginalizes over uncertainties in the mapping between galaxies and DM halos, the properties of the MW system, and the disruption of subhalos by the MW disk. Our results are consistent with the cold, collisionless DM paradigm and yield the strongest cosmological constraints to date on particle models of warm, interacting, and fuzzy dark matter. At $95\%$ confidence, we report limits on ($\mathit{i}$) the mass of thermal relic warm DM, $m_{\rm WDM} > 6.5\ \mathrm{keV}$ (free-streaming length, $\lambda_{\rm{fs}} \lesssim 10\,h^{-1}\ \mathrm{kpc}$), ($\mathit{ii}$) the velocity-independent DM$-$proton scattering cross section, $\sigma_{0} < 8.8\times 10^{-29}\ \mathrm{cm}^{2}$ for a $100\ \mathrm{MeV}$ DM particle mass (DM$-$proton coupling, $c_p \lesssim (0.3\ \mathrm{GeV})^{-2}$), and ($\mathit{iii}$) the mass of fuzzy DM, $m_{\phi}> 2.9 \times 10^{-21}\ \mathrm{eV}$ (de Broglie wavelength, $\lambda_{\rm{dB}} \lesssim 0.5\ \mathrm{kpc}$). These constraints are complementary to other observational and laboratory constraints on DM properties. Show Partial Abstract