Three topics in additive prime number theory

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.