We survey the landscape of binary hydrides across the entire periodic table from 10 to 500 GPa using a crystal structure prediction method. Building a critical temperature ($T_c$) model, with inputs arising from density of states calculations and Gaspari-Gyorffy theory, allows us to predict which energetically competitive candidates are most promising for high-$T_c$ superconductivity. Implementing optimisations, which lead to an order of magnitude speed-up for electron-phonon calculations, then allows us to perform an unprecedented number of high-throughput calculations of $T_c$ based on these predictions and to refine the model in an iterative manner. Converged electron-phonon calculations are performed for 121 of the best candidates from the final model. From these, we identify 36 above-100 K dynamically stable superconductors. To the best of our knowledge, superconductivity has not been previously studied in 27 of these. Of the 36, 18 exhibit superconductivity above 200 K, including structures of NaH$_6$ (248-279 K) and CaH$_6$ (216-253 K) at the relatively low pressure of 100 GPa.
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