A few months ago, Yitang Zhang announced that there are infinitely many pairs of primes, separated by a distance no more than 70000000. This initiated an extensive collaborative effort (known as polymath8) to reduce this bound by optimising different parts of the proof, until eventually reaching a bound of about 5000. This seemed to be the limit of sieve methods, and interest in this area dried up slightly.
However, recently Terry Tao and James Maynard independently developed a new method (also exploiting short admissible k0-tuples), reducing the bound to 600. This, whilst being much closer to the conjectured bound of 2, is not the most exciting result proved by Tao and Maynard. Indeed, they have shown that for any m, we can find H(m) such that there exist infinitely many intervals of width H(m) containing m + 1 primes.
The research is detailed in Andrew Granville’s paper on the topic.