Multi-messenger astrophysics is a novel discipline aiming at unique insights into the properties of cosmic sources and emission processes at work in the Universe. These insights arise from the complementary information carried by the diverse cosmic messengers, namely photons, gravitational waves, neutrinos and cosmic rays. Opening new observational windows onto the Universe beyond tradiational astronomy has presented enormous challenges. The scientific yield has already been substantial, with the discovery of high energy cosmic neutrinos in 2013 and the first observation of gravitational waves in 2015. In this colloquium, we survey the current status of high-energy neutrino astrophysics, highlighting observational methods and breakthrough recent results. Key recent achievements include the identification of two distinctive classes of potential neutrino and multi-messenger emitters as well as the first observation of the Milky Way with messengers other than photons. We then outline questions these results have raised a vision of possible future opportunities for this rapidly advancing field.