The design and construction of CARMENES has been presented at previous SPIE conferences. It is a next-generation radial-velocity instrument at the 3.5m telescope of the Calar Alto Observatory, which was built by a consortium of eleven Spanish and German institutions. CARMENES consists of two separate échelle spectrographs covering the wavelength range from 0.52 to 1.71μm at a spec-tral resolution of R < 80,000, fed by fibers from the Cassegrain focus of the telescope. CARMENES saw "First Light" on Nov 9, 2015. During the commissioning and initial operation phases, we established basic performance data such as throughput and spectral resolution. We found that our hollow-cathode lamps are suitable for precise wavelength calibration, but their spectra contain a number of lines of neon or argon that are so bright that the lamps cannot be used in simultaneous exposures with stars. We have therefore adopted a calibration procedure that uses simultaneous star / Fabry Pérot etalon exposures in combination with a cross-calibration between the etalons and hollow-cathode lamps during daytime. With this strategy it has been possible to achieve 1-2 m/s precision in the visible and 5-10 m/s precision in the near-IR; further improvements are expected from ongoing work on temperature control, calibration procedures and data reduction. Comparing the RV precision achieved in different wavelength bands, we find a "sweet spot" between 0.7 and 0.8μm, where deep TiO bands provide rich RV information in mid-M dwarfs. This is in contrast to our pre-survey models, which predicted comparatively better performance in the near-IR around 1μm, and explains in part why our near-IR RVs do not reach the same precision level as those taken with the visible spectrograph. We are now conducting a large survey of 340 nearby M dwarfs (with an average distance of only 12pc), with the goal of finding terrestrial planets in their habitable zones. We have detected the signatures of several previously known or suspected planets and also discovered several new planets. We find that the radial velocity periodograms of many M dwarfs show several significant peaks. The development of robust methods to distinguish planet signatures from activity-induced radial velocity jitter is therefore among our priorities. Due to its large wavelength coverage, the CARMENES survey is generating a unique data set for studies of M star atmospheres, rotation, and activity. The spectra cover important diagnostic lines for activity (H alpha, Na I D1 and D2, and the Ca II infrared triplet), as well as FeH lines, from which the magnetic field can be inferred. Correlating the time series of these features with each other, and with wavelength-dependent radial velocities, provides excellent handles for the discrimination between planetary companions and stellar radial velocity jitter. These data are also generating new insight into the physical properties of M dwarf atmospheres, and the impact of activity and flares on the habitability of M star planets.