In his paper "Bounded gaps between primes", Yitang Zhang proves that there are infinitely many pairs of prime numbers which differ by less than $70,000,000$. Which is currently the best improvement on this result? Also, where can I find a list of all the results that have been obtained as regards Zhang's theorem (I guess we have to check the Polymath Project)?
I don't follow the precise status. But I think you can do worse than keeping an eye on the blog of Terence tao http://terrytao.wordpress.com/ and his posts on the Polymath 8b project.
This is the page which keeps the current best bounds, with a timeline of the bounds being linked to.
I don't know where you can find a comprehensive list of all the results obtained from it but the papers by Pintz and Maynard are the big ones.