Using the available observations in 1699, 1865-1866, and 1965, an orbit has been recomputed for periodic comet 55P/Tempel-Tuttle, the parent body of the Leonid meteors. The comet's motion was numerically integrated back in time for two millennia and ephemerides were computed for each perihelion return. Chinese observations of the comet in 1366 are well represented, and possible (but not definite) observations of the comet in October 1234 and January 1035 have been identified. The Leonid meteors have been observed sinceAD902. Prior to the eighth century, no Leonid meteors should have been visible because the comet's orbit passed too far outside that of the Earth. By using previously recorded accounts of the Leonid meteor showers and storms as a guide, predictions for the forthcoming Leonid displays in November of 1996-1999 are provided. The 1998 and 1999 events hold the most promise for an impressive Leonid meteor display, although there seems little likelihood that the great storms of 1833 and 1966 will be repeated at this apparition.