Abstract: On of the most famous problems in number theory is to show that there are arbitrarily long arithmetic progressions, all of whose terms are prime numbers. At the start of the year, this was known to be true only for 3-term progressions. Recently, Ben Green and Terrence Tao have released a manuscript which purports to prove this conjecture. I will discuss some of the history of this problem, and outline their argument. I intend for this talk to be accessible to a broad audience, and hope that this will lead to a series of talks (not all by me) verifying the Green/Tao proof.