The Zwicky Transient Facility (ZTF) is a new robotic time-domain survey currently in progress using the Palomar 48-inch Schmidt Telescope. ZTF uses a 47 square degree field with a 600 megapixel camera to scan the entire northern visible sky at rates of ~3760 square degrees/hour to median depths of g ~ 20.8 and r ~ 20.6 mag (AB, 5sigma in 30 sec). We describe the Science Data System that is housed at IPAC, Caltech. This comprises the data-processing pipelines, alert production system, data archive, and user interfaces for accessing and analyzing the products. The realtime pipeline employs a novel image-differencing algorithm, optimized for the detection of point source transient events. These events are vetted for reliability using a machine-learned classifier and combined with contextual information to generate data-rich alert packets. The packets become available for distribution typically within 13 minutes (95th percentile) of observation. Detected events are also linked to generate candidate moving-object tracks using a novel algorithm. Objects that move fast enough to streak in the individual exposures are also extracted and vetted. The reconstructed astrometric accuracy per science image with respect to Gaia is typically 45 to 85 milliarcsec. This is the RMS per axis on the sky for sources extracted with photometric S/N >= 10. The derived photometric precision (repeatability) at bright unsaturated fluxes varies between 8 and 25 millimag. Photometric calibration accuracy with respect to Pan-STARRS1 is generally better than 2%. The products support a broad range of scientific applications: fast and young supernovae, rare flux transients, variable stars, eclipsing binaries, variability from active galactic nuclei, counterparts to gravitational wave sources, a more complete census of Type Ia supernovae, and Solar System objects.