  We introduce a refinement of the GPY sieve method for studying prime
$k$-tuples and small gaps between primes. This refinement avoids previous
limitations of the method, and allows us to show that for each $k$, the prime
$k$-tuples conjecture holds for a positive proportion of admissible $k$-tuples.
In particular, $\liminf_{n}(p_{n+m}-p_n)<\infty$ for any integer $m$. We also
show that $\liminf(p_{n+1}-p_n)\le 600$, and, if we assume the
Elliott-Halberstam conjecture, that $\liminf_n(p_{n+1}-p_n)\le 12$ and
$\liminf_n (p_{n+2}-p_n)\le 600$.
