© 2016. The American Astronomical Society. All rights reserved.. 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 3622-10354 A and an average footprint of ∼500 arcsec2 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 algorithms and centralized metadata framework that produce sky-subtracted spectrophotometrically calibrated spectra and rectified three-dimensional 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, we demonstrate that the MaNGA data have nearly Poisson-limited sky subtraction shortward of ∼8500 A and reach a typical 10σ limiting continuum surface brightness μ = 23.5 AB arcsec-2 in a five-arcsecond-diameter aperture in the g-band. The wavelength calibration of the MaNGA data is accurate to 5 km s-1 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 σ = 72 km s-1.