High-energy neutrinos could be produced in the interaction of charged cosmic rays with matter or radiation surrounding astrophysical sources. Even with the recent detection of extraterrestrial high-energy neutrinos by the IceCube experiment, no astrophysical neutrino source has yet been discovered. Transient sources, such as gamma-ray bursts, core-collapse supernovae, or active galactic nuclei are promising candidates. Multi-messenger programs offer a unique opportunity to detect these transient sources. By combining the information provided by the ANTARES neutrino telescope with information coming from other observatories, the probability of detecting a source is enhanced, allowing the possibility of identifying a neutrino progenitor from a single detected event. A method based on optical and X-ray follow-ups of high-energy neutrino alerts has been developed within the ANTARES collaboration. This method does not require any assumptions on the relation between neutrino and photon spectra other than time-correlation. This program, denoted as TAToO, triggers a network of robotic optical telescopes (TAROT and ROTSE) and the Swift-XRT with a delay of only a few seconds after a neutrino detection, and is therefore well-suited to search for fast transient sources. To identify an optical or Xray counterpart to a neutrino signal, the images provided by the follow-up observations are analysed with dedicated pipelines. A total of 42 alerts with optical and 7 alerts with Xray images taken with a maximum delay of 24 hours after the neutrino trigger have been analysed. No optical or X-ray counterparts associated to the neutrino triggers have been found, and upper limits on transient source magnitudes have been derived. The probability to reject the gamma-ray burst origin hypothesis has been computed for each alert.