Many of the astrophysical sources and violent phenomena observed in our Universe are potential emitters of gravitational waves and high-energy cosmic radiation, including photons, hadrons, and presumably also neutrinos. Both gravitational waves (GW) and high-energy neutrinos (HEN) are cosmic messengers that may escape much denser media than photons. They travel unaffected over cosmological distances, carrying information from the inner regions of the astrophysical engines from which they are emitted (and from which photons and charged cosmic rays cannot reach us). For the same reasons, such messengers could also reveal new, hidden sources that have not been observed by conventional photon-based astronomy. Coincident observation of GWs and HENs may thus play a critical role in multimessenger astronomy. This is particularly true at the present time owing to the advent of a new generation of dedicated detectors: the neutrino telescopes IceCube at the South Pole and ANTARES in the Mediterranean Sea, as well as the GW interferometers Virgo in Italy and LIGO in the United States. Starting from 2007, several periods of concomitant data taking involving these detectors have been conducted. More joint data sets are expected with the next generation of advanced detectors that are to be operational by 2015, with other detectors, such as KAGRA in Japan, joining in the future. Combining information from these independent detectors can provide original ways of constraining the physical processes driving the sources and also help confirm the astrophysical origin of a GW or HEN signal in case of coincident observation. Given the complexity of the instruments, a successful joint analysis of this combined GW and HEN observational data set will be possible only if the expertise and knowledge of the data is shared between the two communities. This Colloquium aims at providing an overview of both theoretical and experimental state of the art and perspectives for GW and HEN multimessenger astronomy.