The n-body problem
The n-body problem has seen ever since Newton a multitude of attempts to solve, but it showed shortly before the 20th century that there was no closed solution in mathematical formulae. Chronolocigally only some stages.
After Newton nearly all great mathematicians of their times tried to solve the problem: Clairaut, Euler, Lagrange, d'Alembert, Laplace, Gauß, Legendre and many many more... All names where everyone involved in mathematics knows the importance and what big contributions they made to the evolution of mathematics. As an example the french mathematician Clairaut (1713-1765) said: "may solve this integral calculus who ever is able to solve it. I formulated the equations without any problem, but I have tried litle to solve them, because they seem unsolvable to me" ("Intègre maintenant qui pourra! J'ai trouvé les six équations que je viens de trouver dès les premiers temps que j'ai envisagé le problème des trois corps, mais je n'ai jamais fait que peux d'efforts pour les résoudre parce qu'elles m'ont toujours paru peu traitables.".).
It is absolutely no wonder that french mathematicians contributet the biggest advances at this time: it was the time of the enlightenment (German "Aufklärung"), of the french revolution(1789), a revolution against all not legitimate authorithies, against feudalism. The biggest impact on european science and the driving force on european quest for knowledge. The metric measurement system was introcuded in many european countries in this time (as a positive side effect of Napoleon's wars), which gave science a tremendous push forward.
It was late in 1888 that Bruns could show that the problem did not lend itself to a general solution in closed form, it was not integratable. French mathematician Pointcaré could on the other hand around 1890 prove for a special case (Bruns & Pointcaré), that the problem was solvable by convergent series expansion.
Numerical solutions were then tried by different mathematicians in 20th century. But even here there showed up unsurmountable problems. Even simplifications as the "problème restreint" (mass of one body very much less than the other two) did not lean itself to a closed solution. Even with a rotating coordinate system that simplified the calculations drastically, no solutions were found. Lagrange had very early introduced the so called "Lagrange points L1-L5" which allowed statements to the stability in these points. But all numerical solutions had stability problems(ejection or capture). Typical numerical problems that you can study and repeat today with computers if the calculations per time unit are too low or if the precission of your computation is not high enough - ie ejection of one body or capture of the body - complicated even these calculations.
The whole 20th century has seen many different attempts to solve the problem from different starting points which I don't enumerate here. If you're interested in these, there is enough literature you can read, maybe you even find something in the net.
Some more remarks to this theme you find under 'Site history' and to the history of astronomy under History of astronomy.
http://www.scholarpedia.org/article/Three_body_problem (short and main details)
http://www.gap-system.org/~history/PrintHT/Orbits.html or http://www-history.mcs.st-and.ac.uk/HistTopics/Orbits.html