  Given a graph $H$, the Ramsey number $r(H)$ is the smallest natural number
$N$ such that any two-colouring of the edges of $K_N$ contains a monochromatic
copy of $H$. The existence of these numbers has been known since 1930 but their
quantitative behaviour is still not well understood. Even so, there has been a
great deal of recent progress on the study of Ramsey numbers and their
variants, spurred on by the many advances across extremal combinatorics. In
this survey, we will describe some of this progress.
