Two months after the discovery of comet Shoemaker-Levy 9 came the astonishing announcement that the comet would impact Jupiter in July 1994. Computing the orbital motion of this remarkable comet presented several unusual challenges. We review the pre-impact orbit computations and impact predictions for SL9, from the preliminary orbit solutions shortly after discovery to the final set of predictions before the impacts. The final set of predicted impact times were systematically early by an average of 7 minutes, probably due to systematic errors in the reference star catalogs used in the reduction of the fragments' astrometric positions. The actual impact times were inferred from the times of observed phenomena for 16 of the impacts. Orbit solutions for the fragments were refined by using the actual impact times as additional data, and by estimating and removing measurement biases from the astrometric observations. The final orbit solutions for 21 fragments are tabulated, along with final estimates of the impact times and locations. The pre-breakup orbital history of the comet was investigated statistically, via a Monte Carlo analysis. The progenitor nucleus of SL9 was most likely captured by Jupiter around 1929 ± 9 years. Prior to capture, the comet was in a low-eccentricity, low-inclination heliocentric orbit entirely inside Jupiter's orbit, or, less likely, entirely outside. The ensemble of possible pre-capture orbits is consistent with a group of Jupiter family comets known as the quasi-Hildas.