The Mu2e experiment at Fermilab plans to search for charged-lepton flavor violations by looking for neutrino-less muon to electron conversions in the field of the nucleus. A complex solenoid system and precise knowledge of its magnetic field play a major role in the experimental approach Mu2e has chosen. It is essential to map the solenoid field up to 10 -4 accuracy. This article describes the design of the Field Mapping System Mu2e will use to measure the magnetic field. Two different mechanical mapper systems, a survey-based position determination of the in-house calibrated three-dimensional Hall probes, a motion control system, and a data acquisition and readout system are presented.