Here is the basic animation of the figure-eight configuration (using Carlos Simó's data):

The motion of the three bodies looks plausible as the track of a system of three bodies interacting through Newton's law of gravitation. But this could be an illusion. There is much numerical evidence that this is indeed a good picture of a real three-body system. But numerical evidence cannot distinguish between a real periodic motion of three bodies and one that very closely approximates this one for a while and then diverges away from it. The current mathematical interest of this topic is that Chenciner and Montgomery have recently proven that there is a choreographic system that has many features in common with the orbit suggested by computation and is in fact plausibly approximated by the techniques that produced the picture above.