Interest in hydrogen as a future energy carrier in mobile applications has led to a strong increase in research on the structural properties of complex alkali metal and alkaline earth hydrides, with the aim to find structural phases with higher hydrogen densities. This contribution reviews recent work on the structural properties and phase diagrams of these complex hydrides under elevated pressures, an area where rapid progress has been made over the last few years. The materials discussed in greatest detail are LiAlH4, NaAlH4, Li3AlH6, Na3AlH6, LiBH4, NaBH4, and KBH4. All of these have been studied under high pressure by various methods such as X-ray or neutron scattering, Raman spectroscopy, differential thermal analysis or thermal conductivity measurements in order to find information on their structural phase diagrams. Based mainly on experimental studies, preliminary or partial phase diagrams are also given for six of these materials. In addition to this information, data are provided also on experimental results for a number of other complex hydrides, and theoretical predictions of new phases and structures under high pressures are reviewed for several materials not yet studied experimentally under high pressure.