One century after the seminal balloon flights of Victor Hess, the Pierre Auger Observatory aims at unveiling some of the mysteries of the highest-energy cosmic rays: what are their sources? Is there an end to the spectrum? What kind of particles are they? Are there signatures of new physics or of a violation of fundamental laws of physics? The Auger Observatory measures cosmic rays with energies of 10^20 eV or more by observing the giant air showers created when the particles hit the atmosphere. Located in Argentina, two complementary detector systems are used: an array of 1600 water-Cherenkov detectors distributed over 3000 sqkm, and fluorescence telescopes which monitor the atmosphere above the array in clear nights. Since 2005, data of unprecedented quantity and quality could be taken. In the talk, the observation principles, successes and limitations are described. Current, partly surprising results are presented. Data interpretations related to the search for violation of Lorentz invariance are mentioned.