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Material from the surface of a planet can be ejected into space by a large impact and could carry primitive life-forms with it. We performed n-body simulations of such ejecta to determine where in the Solar System rock from Earth and Mars may end up. We found that, in addition to frequent transfer of material among the terrestrial planets, transfer of material from Earth and Mars to the moons of Jupiter and Saturn is also possible, but rare. We expect that such transfers were most likely to occur during the Late Heavy Bombardment or during the ensuing 1-2 billion years. At this time, the icy moons were warmer and likely had little or no ice shell to prevent meteorites from reaching their liquid interiors. We also note significant rates of re-impact in the first million years after ejection. This could re-seed life on a planet after partial or complete sterilization by a large impact, which would aid the survival of early life during the Late Heavy Bombardment. Key Words: Panspermia-Impact-Meteorites-Titan-Europa. Astrobiology 13, 1155-1165. [ABSTRACT FROM AUTHOR]