Big Doings from Small Gaps

In the past two years, the mathematical community has been stunned by one breakthrough after another in the study of the distribution of prime numbers. The twin prime conjecture, long considered a "holy grail", seems now almost within reach. This has prompted mathematicians to ask which other questions about primes can be approached by the powerful new methods of Zhang, Maynard, and Tao. It seems safe to say that this project of reassessment will continue for some time. We survey many exciting applications that have been found so far, and we discuss some of what still remains to be done.