The Imaging Magnetograph eXperiment (IMaX) is a spectropolarimeter built by four institutions in Spain that flew on board the Sunrise balloon-borne solar observatory in June 2009 for almost six days over the Arctic Circle. As a polarimeter, IMaX uses fast polarization modulation (based on the use of two liquid crystal retarders), real-time image accumulation, and dual-beam polarimetry to reach polarization sensitivities of 0.1%. As a spectrograph, the instrument uses a LiNbO3 etalon in double pass and a narrow band pre-filter to achieve a spectral resolution of 85 mÅ. IMaX uses the high-Zeeman-sensitive line of Fe i at 5250.2 Å and observes all four Stokes parameters at various points inside the spectral line. This allows vector magnetograms, Dopplergrams, and intensity frames to be produced that, after reconstruction, reach spatial resolutions in the 0.15 - 0.18 arcsec range over a 50×50 arcsec field of view. Time cadences vary between 10 and 33 s, although the shortest one only includes longitudinal polarimetry. The spectral line is sampled in various ways depending on the applied observing mode, from just two points inside the line to 11 of them. All observing modes include one extra wavelength point in the nearby continuum. Gauss equivalent sensitivities are 4 G for longitudinal fields and 80 G for transverse fields per wavelength sample. The line-of-sight velocities are estimated with statistical errors of the order of 5 - 40 m s-1. The design, calibration, and integration phases of the instrument, together with the implemented data reduction scheme, are described in some detail.