High order multistep methods, run at constant stepsize, are one of the most effective schemes for integrating the Newtonian solar system, for long periods of time. I have studied the stability and error growth of these methods, when applied to harmonic oscillators and two-body systems like the Sun-Jupiter pair. I have found that the truncation error of multistep methods on two-body systems grows in time like t^2, and the roundoff like t^{1.5} and I have a theory that accounts for this. I have also tried to design better multistep integrators than the traditional Stormer and Cowell methods, and have found a few interesting ones. A second result of my search for new methods is that I did not find any predictor that is stable on the Sun-Jupiter system, for stepsizes smaller than 108 steps per cycle, whose order of accuracy is greater than 12. For example, Stormer-13 becomes unstable at 108 steps per cycle. This limitation between stability and accuracy seems to be a general property of multistep methods.