The two-body problem is simple; its solution is that each body travels along a conic section which has a focus at the centre of mass of the system.
The three-body problem is much more complicated; its solution can be chaotic. The general three-body problem has not been solved analytically, although approximate solutions can be calculated by numerical methods or perturbation methods.
A restricted three-body problem, in which two of the bodies are in circular orbits and the third is of negligible mass (approximated by the Sun - Earth - Moon system) was solved analytically by Lagrange in the 18th century. The points where the smallest object can orbit the other two with the same period are called Lagrangian points.