A search for cosmic neutrino sources using six years of data collected by the ANTARES neutrino telescope has been performed. Clusters of muon neutrinos over the expected atmospheric background have been looked for. No clear signal has been found. The most signal-like accumulation of events is located at equatorial coordinates R.A. = -46 degrees.8 and decl. = -64 degrees.9 and corresponds to a 2.2 sigma background fluctuation. In addition, upper limits on the flux normalization of an E-2 muon neutrino energy spectrum have been set for 50 pre-selected astrophysical objects. Finally, motivated by an accumulation of seven events relatively close to the Galactic Center in the recently reported neutrino sample of the IceCube telescope, a search for point sources in a broad region around this accumulation has been carried out. No indication of a neutrino signal has been found in the ANTARES data and upper limits on the flux normalization of an E-2 energy spectrum of neutrinos from point sources in that region have been set. The 90% confidence level upper limits on the muon neutrino flux normalization vary between 3.5 and 5.1 x 10(-8) GeV cm(-2) s(-1), depending on the exact location of the source.