  Over the past decade, a combination of crystal structure prediction
techniques and experimental synthetic work has thoroughly explored the phase
diagrams of binary hydrides under pressure. The fruitfulness of this dual
approach is demonstrated in the recent identification of several
superconducting hydrides with $T_c$s approaching room temperature. We start
with an overview of the computational procedures for predicting stable
structures and estimating their propensity for superconductivity. A survey of
phases with high $T_c$ reveals some common structural features that appear
conducive to the strong coupling of the electronic structure with atomic
vibrations that leads to superconductivity. We discuss the stability and
superconducting properties of phases containing two of these -- molecular H$_2$
units mixed with atomic H and hydrogenic clathrate-like cages -- as well as
more unique motifs. Finally, we argue that ternary hydride phases, which are
far less-explored, are a promising route to achieving simultaneously
superconductivity at high temperatures and stability at low pressures. Several
ternary hydrides arise from the addition of a third element to a known binary
hydride structure through site mixing or onto a new site -- and several more
are based on altogether new structural motifs.
