The Pierre Auger Observatory, located in Argentina, has been detecting ultra-high energy cosmic rays for more than ten years. An essential feature of the 3000 km$^2$ Observatory is its hybrid design: cosmic rays above $\simeq 10^{17}~$eV are detected through the observation of the extensive air showers they induce in the atmosphere with complementary techniques, from surface detector arrays and fluorescence telescopes to radio antennas. The analyses of the multi-detector data have enabled high-statistics and high-precision studies of the energy spectrum, mass composition and distribution of arrival directions of Ultra-High Energy Cosmic Rays (UHECRs), which are the main observables to study the origin of these particles. The data have also enabled a series of analyses complementary to the primary objectives, from the search for ultra-high energy photons and neutrinos to the measurement of the proton-air cross section at 57~TeV center-of-mass energy. A coherent interpretation of all these recent results, requiring the use of LHC-tuned interaction models to describe the air showers, opens new directions regarding the properties of UHECR sources. The resulting picture, calling into question the perception of UHECRs at the time of the conception of the Observatory in the early 1990s, has led to a plan for an upgrade of the Observatory.