The results of numerical experiments on the evolution of orbits are used to examine several ideas that comets originate within the solar system. It is shown that comets whose entire orbit lies beyond Jupiter can cross the Jupiter barrier with the help of Saturn's perturbations. Crossing a barrier depends on the current value of the Jacobi quantity. This quantity fluctuates widely during the evolution of individual orbits and cannot be used as an invarient in discussing evolution of comets, asteroids, and meteorites. The numerical experiments show that there must be many comets in Trojan orbits, horseshoe orbits, and mid-range orbits between Jupiter and Saturn. Many orbits starting elsewhere are seen to evolve into these quasi-stable patterns, so it does not necessarily follow that comets originate in these places. If one accepts the hypothesis that comets are formed in the Jupiter-Saturn region, then the orbital evolution is seen to account for the properties of the short-period comets. However, it is shown that comets with such origins cannot become long-period comets. A difficulty with the hypothesis is that two sources are required, one for short-period comets and one for long-period comets. This is contrasted with the alternative hypothesis that all comets approach the solar system initially on near- parabolic orbits. This latter hypothesis is capable of explaining the long-period comets immediately and the short-period comets by evolution.