(Abridged) Blue (BP) and Red (RP) Photometer low-resolution spectral data is one of the exciting new products in Gaia Data Release 3 (Gaia DR3). We calibrate about 65 billion individual transit spectra onto the same mean BP/RP instrument through a series of calibration steps, including background subtraction, calibration of the CCD geometry and an iterative procedure for the calibration of CCD efficiency as well as variations of the line-spread function and dispersion across the focal plane and in time. The calibrated transit spectra are then combined for each source in terms of an expansion into continuous basis functions. Time-averaged mean spectra covering the optical to near-infrared wavelength range 330, 1050 nm are published for approximately 220 million objects. Most of these are brighter than G = 17.65 but some BP/RP spectra are published for sources down to G = 21.43. Their signal- to-noise ratio varies significantly over the wavelength range covered and with magnitude and colour of the observed objects, with sources around G = 15 having S/N above 100 in some wavelength ranges. The top-quality BP/RP spectra are achieved for sources with magnitudes 9 < G < 12, having S/N reaching 1000 in the central part of the RP wavelength range. Scientific validation suggests that the internal calibration was generally successful. However, there is some evidence for imperfect calibrations at the bright end G < 11, where calibrated BP/RP spectra can exhibit systematic flux variations that exceed their estimated flux uncertainties. We also report that due to long-range noise correlations, BP/RP spectra can exhibit wiggles when sampled in pseudo-wavelength.