The Jefferson Lab Electron Ion Collider (JLEIC) is a proposed new machine for a nuclear physics research. As proposed, it will deliver between 15 and 65 GeV center of mass energy collisions between electrons and nuclei. It uses the existing CEBAF accelerator as a full energy injector to deliver 3-10 GeV electrons into a new electron collider ring. An all new ion accelerator and collider complex will deliver up to 100 GeV protons, or 40 GeV/nucleon ions. The machine will have luminosity goals of above 10 33 cm -2 sec -1 in the whole energy range. The two collider rings use a unique figure eight layout to deliver a high degree of polarization in both beams. The machine will ultimately include room for two interaction regions. The crossing angle in the primary interaction region will be 50 mrad. This paper will describe the requirements and preliminary designs for both the ion and electron beam transport systems in the most complex 32 m region around the interaction point. The beam transport system includes three separate cryostats housing over 37 superconducting magnets operating at 4.5 K. The design of these cryostats must be closely integrated with the elements of the JLEIC detector.