Mapping Nearby Galaxies at Apache Point Observatory (MaNGA) is an optical fiber-bundle integral-field unit (IFU) spectroscopic survey that is one of three core programs in the fourth-generation Sloan Digital Sky Survey (SDSS-IV). With a spectral coverage of 3622 - 10,354 Angstroms and an average footprint of ~ 500 arcsec^2 per IFU the scientific data products derived from MaNGA will permit exploration of the internal structure of a statistically large sample of 10,000 low redshift galaxies in unprecedented detail. Comprising 174 individually pluggable science and calibration IFUs with a near-constant data stream, MaNGA is expected to obtain ~ 100 million raw-frame spectra and ~ 10 million reduced galaxy spectra over the six-year lifetime of the survey. In this contribution, we describe the MaNGA Data Reduction Pipeline (DRP) algorithms and centralized metadata framework that produces sky-subtracted, spectrophotometrically calibrated spectra and rectified 3-D data cubes that combine individual dithered observations. For the 1390 galaxy data cubes released in Summer 2016 as part of SDSS-IV Data Release 13 (DR13), we demonstrate that the MaNGA data have nearly Poisson-limited sky subtraction shortward of ~ 8500 Angstroms and reach a typical 10-sigma limiting continuum surface brightness mu = 23.5 AB/arcsec^2 in a five arcsec diameter aperture in the g band. The wavelength calibration of the MaNGA data is accurate to 5 km/s rms, with a median spatial resolution of 2.54 arcsec FWHM (1.8 kpc at the median redshift of 0.037) and a median spectral resolution of sigma = 72 km/s.
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