With the observation of high-energy astrophysical neutrinos, a new era in astroparticle physics has begun. The IceCube Neutrino Telescope has observed neutrinos with unprecedented energies from the depth of the Universe. Observing these neutrinos not only push forward our understanding of the mysterious cosmic particle accelerators but also open up new opportunities to search for phenomena beyond the standard model of particle physics. Very rare processes occurring on the long journeys of energetic neutrinos from their sources to Earth, can be tested, and energy scales beyond the reach of accelerators or high-intensity anthropogenic neutrino beams, can be explored. This talk will focus on the search for new phenomena with the IceCube Neutrino Telescope and the potential for discoveries with the next generation of neutrino detectors.