A system of 5020 robotic fiber positioners was installed in 2019 on the Mayall Telescope, at Kitt Peak National Observatory. The robots automatically retarget their optical fibers every 10-20 minutes, each to a precision of several microns, with a reconfiguration time of fewer than 2 minutes. Over the next 5 yr, they will enable the newly constructed Dark Energy Spectroscopic Instrument (DESI) to measure the spectra of 35 million galaxies and quasars. DESI will produce the largest 3D map of the universe to date and measure the expansion history of the cosmos. In addition to the 5020 robotic positioners and optical fibers, DESI's Focal Plane System includes six guide cameras, four wave front cameras, 123 fiducial point sources, and a metrology camera mounted at the primary mirror. The system also includes associated structural, thermal, and electrical systems. In all, it contains over 675,000 individual parts. We discuss the design, construction, quality control, and integration of all these components. We include a summary of the key requirements, the review and acceptance process, on-sky validations of requirements, and lessons learned for future multiobject, fiber-fed spectrographs.