The lunar geological record is expected to contain a rich record of the Solar System's galactic environment, including records of nearby (i.e. ≤ a few tens of parsecs) supernova explosions. This record will be composed of two principal components: (i) cosmogenic nuclei produced within, as well as radiation damage to, surface materials caused by increases in the galactic cosmic ray flux resulting from nearby supernovae; and (ii) the direct collection of supernova ejecta, likely enriched in a range of unusual and diagnostic isotopes, on the lunar surface. Both aspects of this potentially very valuable astrophysical archive will be best preserved in currently buried, but nevertheless near-surface, layers that were directly exposed to the space environment at known times in the past and for known durations. Suitable geological formations certainly exist on the Moon, but accessing them will require a greatly expanded programme of lunar exploration.