The Pierre Auger Observatory is the largest ultra-high energy cosmic-ray detector built so far in the world. With the Surface Detector array of the Observatory we can also detect ultra-high energy neutrinos with energies around 100 PeV and above. The identification is efficiently done for neutrinos of all flavours interacting in the atmosphere at large zenith angles, as well as for tau neutrinos interacting in the Earth’s crust, covering sky directions with equatorial declinations from about -80° to +60°. The sensitivity obtained summing up all these channels is shown to be comparable to other neutrino detectors in operation, and to constrain several models of cosmic ray and neutrino production in the EeV region. In the absence of candidates in data from 1 January2004 to 31 March 2017, improved stringent limits to diffuse and point source fluxes of ultra-high energy neutrinos are obtained. In addition, we report on the results of the targeted search for neutrinos in temporal and spatial coincidence with the gravitational wave events detected by the Advanced LIGO detectors, GW150914, GW151226 and GW170104, and on their implications for the total energy radiated in EeV neutrinos by identified sources of gravitational waves.