The deexcitation or quenching of the metastable $2S$ state of atomic hydrogen in collision with atoms and molecules has been studied using a beam-attenuation method in conjunction with a time-of-flight technique at velocities between 0.4 \ifmmode\times\else\texttimes\fi{} ${10}^{6}$ and 4 \ifmmode\times\else\texttimes\fi{} ${10}^{6}$ cm/sec (0.08 and 8 eV). In this regime, transfer of the metastable to the $2P$ state of hydrogen, followed by radiative decay to the ground state, is the dominant destruction mechanism. Absolute cross sections are reported for the quenching of $\mathrm{H}(2S)$ atoms in collision with the noble gases (helium-xenon), with molecules that have permanent electric-quadrupole moments (hydrogen and nitrogen), and with molecules that have permanent electric-dipole moments (ammonia, methanol, and acetone). For molecules with dipole moments, the cross sections are on the order of ${10}^{\ensuremath{-}13}$ ${\mathrm{cm}}^{2}$ and vary approximately as ${v}^{\ensuremath{-}1}$. For the noble gases and the quadrupole-moment molecules, the cross sections are on the order of ${10}^{\ensuremath{-}14}$ ${\mathrm{cm}}^{2}$ and vary approximately as ${v}^{\ensuremath{-}n}$ where $0.3<n<0.7$. Measurements of the relative cross section for the production of ultraviolet radiation in collision with nitrogen and argon are reported, and the cross sections for the quenching of $\mathrm{H}(2S)$ and $\mathrm{D}(2S)$ in argon are compared. Data for the noble gases indicate that large-angle elastic scattering is probably not responsible for the discrepancy between theory and experiment. The data for molecular hydrogen suggest that short-range forces are important in collisions with molecules possessing a quadrupole moment.