We describe the spectroscopic data processing pipeline of the Dark Energy Spectroscopic Instrument (DESI), which is conducting a redshift survey of about 40 million galaxies and quasars using a purpose-built instrument on the 4 m Mayall Telescope at Kitt Peak National Observatory. The main goal of DESI is to measure with unprecedented precision the expansion history of the universe with the baryon acoustic oscillation technique and the growth rate of structure with redshift space distortions. Ten spectrographs with three cameras each disperse the light from 5000 fibers onto 30 CCDs, covering the near-UV to near-infrared (3600-9800 Å) with a spectral resolution ranging from 2000 to 5000. The DESI data pipeline generates wavelength- and flux-calibrated spectra of all the targets, along with spectroscopic classifications and redshift measurements. Fully processed data from each night are typically available to the DESI collaboration the following morning. We give details about the pipeline's algorithms, and provide performance results on the stability of the optics, the quality of the sky background subtraction, and the precision and accuracy of the instrumental calibration. This pipeline has been used to process the DESI Survey Validation data set, and has exceeded the project's requirements for redshift performance, with high efficiency and a purity greater than 99% for all target classes.