  The classical hypergraph Ramsey number $r_k(s,n)$ is the minimum $N$ such
that for every red-blue coloring of the $k$-tuples of $\{1,\ldots, N\}$, there
are $s$ integers such that every $k$-tuple among them is red, or $n$ integers
such that every $k$-tuple among them is blue. We survey a variety of problems
and results in hypergraph Ramsey theory that have grown out of understanding
the quantitative aspects of $r_k(s,n)$. Our focus is on recent developments and
open problems.
