**Title:** Primes, Sieves, and Prime Gaps

**Abstract:** This is an expository lecture focusing on the recent surprising achievement by Yitang Zhang, proving that there are infinitely many pairs of consecutive primes with bounded gap, as well as describing a completely independent proof by James Maynard. The talk describes the steps, during two centuries, that led to the solution of the problem and some of the new questions that arose from this work.