In gravitational-wave detection, special emphasis is put onto searches that focus on cosmic events detected by other types of astrophysical observatories. The astrophysical triggers, e.g. from '-ray and x-ray satellites, optical telescopes and neutrino observatories, provide a trigger time for analyzing gravitational-wave data coincident with the event. In certain cases the expected frequency range, source energetics, directional and progenitor information are also available. Beyond allowing the recognition of gravitational waveforms with amplitudes closer to the noise floor of the detector, these triggered searches should also lead to rich science results even before the onset of Advanced LIGO. In this paper we provide a broad review of LIGO's astrophysically triggered searches and the sources they target.