Major milestones in the twin prime conjecture

In April 2013 Yitang Zhang announced a proof that there are infinitely many pairs of prime numbers that have a difference of at most 70 million (see Zhang (2014)). Others have since narrowed the gap from 70 million to just 246. Reducing the gap to 2 would prove the twin prime conjecture. The significance is a path to solving an ancient problem that looked hopeless just a few years ago. The popular press has discussed Zhang's result but at a very high level. Zhang (2014) and subsequent papers appearing in number theory journals present only the most recent details and are not really accessible to the nonspecialist. This paper attempts to bridge the gap and present a history of major milestones leading to the current state of the twin prime conjecture written at the level of working mathematicians.