  This is an expository article to accompany my two lectures at the CDM
conference. I have used this an excuse to make public two sets of notes I had
lying around, and also to put together a short reader's guide to some recent
joint work with T.Tao. Contents: 1. An exposition, without much detail, of the
work of Goldston, Pintz and Yildirim on gaps between primes; 2. A detailed
discussion of the work of Mauduit and Rivat establishing that 50 percent of the
primes have odd digit sum when written in base 2; 3. A reader's guide to recent
work of T.Tao and the author on linear equations in primes.
  The sections can be read independently.
