We analyze the distribution of arrival directions of ultra-high-energy cosmic rays recorded at the Pierre Auger Observatory in 10 years of operation. The data set, about three times larger than that used in earlier studies, includes arrival directions with zenith angles up to 80°, thus covering from -90º to +45º in declination. After updating the fraction of events correlating with the active galactic nuclei (AGNs) in the Véron-Cetty and Véron catalog, we subject the arrival directions of the data with energies in excess of 40 EeV to different tests for anisotropy. We search for localized excess fluxes, self-clustering of event directions at angular scales up to 30°, and different threshold energies between 40 and 80 EeV. We then look for correlations of cosmic rays with celestial structures both in the Galaxy (the Galactic Center and Galactic Plane) and in the local universe (the Super-Galactic Plane). We also examine their correlation with different populations of nearby extragalactic objects: galaxies in the 2MRS catalog, AGNs detected by Swift-BAT, radio galaxies with jets, and the Centaurus A (Cen A) galaxy. None of the tests show statistically significant evidence of anisotropy. The strongest departures from isotropy (post-trial probability ~1.4%) are obtained for cosmic rays with E > 58 EeV in rather large windows around Swift AGNs closer than 130 Mpc and brighter than 1044 erg s−1 (18° radius), and around the direction of Cen A (15° radius).