  How long does it take a random searcher to visit all sites of a given domain?
This time, known as the cover time, is a key observable to quantify the
efficiency of exhaustive searches, which require a complete exploration of an
area and not only the discovery of a single target; examples range from immune
system cells chasing pathogens to animals harvesting resources, robotized
exploration by e.g. automated cleaners or deminers, or algorithmics. Despite
its broad relevance, the cover time has remained elusive and so far explicit
results have been scarce and mostly limited to regular random walks. Here we
determine the full distribution of the cover time for a broad range of random
search processes, which includes the prominent examples of L\'evy strategies,
intermittent strategies, persistent random walks and random walks on complex
networks, and reveal its universal features. We show that for all these
examples the mean cover time can be minimized, and that the corresponding
optimal strategies also minimize the mean search time for a single target,
unambiguously pointing towards their robustness.
