Can one solve the equations that describe three mutually attracting bodies revolving around each other in space?

The first three-body problem studied was the one involving the Moon, the Earth, and the Sun.  This classic problem stimulated much of celestial mechanics and much of mathematics too.   One could say that modern physics was born in order to study the three-body problem.

In the time of Newton, the general three-body problem was considered insoluble.  With the birth of general relativity around 1910, the apparently simpler​​ two-body problem was reclassified as​ insoluble.  When quantum electrodynamics came along in 1930, even the one-body problem was deemed​ insoluble.    And in modern quantum field theory, the problem of zero bodies—i.e., the vacuum—“arguably” has no mathematical solution.  It follows that​ if we are seeking an exact description, no bodies at all are already too many!

But in practice, simplifying assumptions can be usually be made and mathematically exact solutions are not needed. Astronomers can make predictions that are good for all practical purposes. Jupiter is not going to crash into Neptune at any time in the foreseeable future.

There is a beautiful solution where three equally massive bodies chase each other forever around a figure-eight.  It would make a nice kinetic mandala for zen contemplation as, in an endless dance, the three bodies periodically switch partners.  This solution is (just barely) stable.  Hence it is possible, although unlikely, that one day astronomers will observe such a configuration somewhere out there in the vast reaches of the universe.