A survey of $\chi$-boundedness

If a graph has bounded clique number, and sufficiently large chromatic number, what can we say about its induced subgraphs? Andr'as Gy'arf'as made a number of challenging conjectures about this in the early 1980’s, which have remained open until recently; but in the last few years there has been substantial progress. This is a survey of where we are now.