It is believed that there should be infinitely many pairs of primes which differ by 2; this is the famous twin prime conjecture. More generally, it is believed that for every positive integer m there should be infinitely many sets of m primes, with each set contained in an interval of size roughly m log{m}. We will introduce a refinement of the GPY sieve method for studying these problems. This refinement will allow us to show (amongst other things) that liminf_n(p_{n+m}-p_n)<∞ for any integer m, and so there are infinitely many bounded length intervals containing m primes.