Where is quantum physics with regards to the periodic table? In his Lecture's on Physics (circa 1960's) Richard Feynman wrote that so far physics has only been able to model (solve) the hydrogen and helium atoms. So now, more than 50 year's later where are we in the periodic table? Have we progressed to heavier elements? If so what element?
And what are (were) the critical hurdles that delay this progress?
 A: That depends on what is meant by "solving" the atom. What Feynman probably is referring to is the usual atomic Hamiltonian, which is already an approximation from the field theoretic point of view (no strong forces, etc.).
The main problem is electron-electron-interactions. If you have an atom with more than one electron, the interaction term between the electrons makes an analytic solution impossible (I guess that because this is essentially a three-body problem, you just can't give analytical solutions, but I can't give a solid reference).
Therefore, you have to do further approximations. There are many ways to do so and often, our computers are strong enough to simulate about anything we want to do, i.e. we know a good approximation to the electron structure of all the elements currently known. 
A good starting point for learning about approximation schemes is Wikipedia. One of the best known methods is the Hartree-Fock approximation, which is essentially a mean field method. Starting there http://en.wikipedia.org/wiki/Hartree%E2%80%93Fock_method, at the side you'll find a whole zoo of different methods to approximate the electronic structure. They are not all about determining the structure of a single atom, but I guess at least some of them will be. 
Note that for larger atoms, the usual non-relativistic quantum mechanics cannot give correct results. This is, of course, just about the electron structure. If you want to ask different questions, you might have to use different methods. 
