We present and interpret observations of atomic and molecular gas toward the southern elliptical galaxy NGC 1316 (Fornax A), a strong double-lobe radio source with a disturbed optical morphology that includes numerous shells and loops. The 12CO(1-0), 12CO(2-1), and HI observations were made with SEST and the VLA. CO emission corresponding to a total molecular hydrogen mass of 5x10^8 Msun was detected toward the central position as well as northwest and southeast of the nucleus in the regions of the dust patches. The origin of that gas is likely external and due to accretion of one or several small gas-rich galaxies. HI was not detected in the central region of NGC1316, but ~2x10^7 Msun of atomic gas was found toward the giant HII region discovered by Schweizer (1980) located 6.7 arcmin (or 36.2 kpc) from the nucleus. HI was also found at three other locations in the outer part of NGC 1316. The HI distributions and kinematics of the two nearby spiral companions of NGC 1316, NGC 1317 (a barred galaxy to the north) and NGC 1310 (to the west) could be studied. Both galaxies have unusually small HI disks that may have been affected by ram-pressure stripping.