Bounded gaps between primes

These are notes on Zhang's work and subsequent developments produced in
preparation for 5 hours of talks for a general mathematical audience given in
Cambridge, Edinburgh and Auckland over the last year.
Being for colloquium-style talks, these notes are at a much lower level than
other accounts in the literature. In places they are deliberately quite
nonrigorous. Whether anyone will find them helpful is unclear but some care was
taken on their preparation and there is, I hope, little harm in making them
available here.