The properties of hot dense hydrogen and helium are important for understanding many of the recently observed giant planets but reliable estimates from either theory or experiment of the properties of hydrogen and helium under appropriate conditions remain a challenge. The big question relevant to planetary interiors is the miscibility of helium in metallic hydrogen; we provide robust estimates which help to resolve the observed differences between Jupiter and Saturn. Another important issue is how hydrogen goes from a molecular insulating phase to an atomic metallic phase. We observe clear evidence of an 'extra' liquid-liquid phase transition as predicted by Zeldovitch and Landau. In looking for the existence of a low temperature liquid of high pressure hydrogen we have found new ground-state crystal structures which are likely to be "high temperature" superconductors.