Abstract: The study of small gaps between prime numbers is motivated by the twin-prime conjecture on the infinitude of primes p such that p+2 is also prime. Recently, D. Goldston, J. Pintz and C. Yildirim achieved a breakthrough in the field and proved results which had previously been considered beyond reach. In this talk, I will survey the history of the problem and some of the techniques used to study it. I will then go on to sketch the Goldston-Pintz-Yildirim method.