JADES Data Release 3 -- NIRSpec/MSA spectroscopy for 4,000 galaxies in the GOODS fields

We present the third data release of JADES, the JWST Advanced Deep Extragalactic Survey, providing both imaging and spectroscopy in the two GOODS fields. Spectroscopy consists of medium-depth and deep NIRSpec/MSA spectra of 4,000 targets, covering the spectral range 0.6-5.3 $\mu$m and observed with both the low-dispersion prism (R=30-300) and all three medium-resolution gratings (R=500-1,500). We describe the observations, data reduction, sample selection, and target allocation. We measured 2,375 redshifts (2,053 from multiple emission lines); our targets span the range from z=0.5 up to z=13, including 404 at z>5. The data release includes 2-d and 1-d fully reduced spectra, with slit-loss corrections and background subtraction optimized for point sources. We also provide redshifts and S/N>5 emission-line flux catalogs for the prism and grating spectra, and concise guidelines on how to use these data products. Alongside spectroscopy, we are also publishing fully calibrated NIRCam imaging, which enables studying the JADES sample with the combined power of imaging and spectroscopy. Together, these data provide the largest statistical sample to date to characterize the properties of galaxy populations in the first billion years after the Big Bang.