The Advanced X ray Astrophysics Facility (AXAF) contains two focal-plane science instruments -- the microchannel-plate high-resolution camera (HRC) and the AXAF CCD imaging spectrometer (ACIS). Each of these instruments provides two low-internal- background detectors, one for imaging and the other for reading out AXAF's objective transmission gratings. Taking maximum advantage of the low internal background of these instruments requires shielding against external electromagnetic and particle radiation. To optimize the shielding within weight constraints, we performed extensive numerical simulations of the photon transport and of the particle transport and activation within the AXAF. We discuss the simulations and report the results of the shielding study. With the adopted shielding design, the particle-induced background and the x-ray-induced background are each comparable to or less than the anticipated internal background of AXAF's imaging detectors (HRC and ACIS). Thus, the predicted total non-imaged background is very low -- less than 0.2 counts arcsec(superscript -2) day(superscript -1) for the HRC and even lower for the ACIS half-arcsec angular resolution, this low background has negligible effect upon point-source detection, for reasonable observation durations.