Neutrinos provide a unique view of the universe, able to travel distances and escape from environments that light cannot. As a result of their low interaction rates with matter, detection of these cosmic messengers requires cubic-kilometer or larger deep underground detectors. Elimination of background events from cosmic rays that are billions of times more common and reconstruction of neutrino events from the very sparse detector arrays required by large volumes and finite funding pose substantial computational challenges. In the case of IceCube, these challenges are magnified by the constraints of working at the South Pole with extremely limited network connectivity and a limited power budget. In this talk, I will discuss the approaches adopted by IceCube and the challenges on the horizon for future larger and higher-rate instruments.