  In the recent preprint [3], Goldston, Pintz, and Y{\i}ld{\i}r{\i}m
established, among other things, $$ \liminf_{n\to\infty}{p_{n+1}-p_n\over\log
p_n}=0,\leqno(0) $$ with $p_n$ the $n$th prime. In the present article, which
is essentially self-contained, we shall develop a simplified account of the
method used in [3]. While [3] also includes quantitative versions of $(0)$, we
are concerned here solely with proving the qualitative $(0)$, which still
exhibits all the essentials of the method. We also show here that an
improvement of the Bombieri--Vinogradov prime number theorem would give rise
infinitely often to bounded differences between consecutive primes. We include
a short expository last section. Detailed discussions of quantitative results
and a historical review will appear in the publication version of [3] and its
continuations.

Number Theory