A graph $G$ is $\chi$-bounded by the function $f$ if every induced subgraph $H$ of $G$ satisfied $\chi(H) \le f(\omega(H))$. A class of graphs is $\chi$-bounded if there exists a function $f$ such that every graph in the class is $\chi$-bounded by $f$. It is polynomially $\chi$-bounded if there is such a function $f$ that is a polynomial. Some classes of graphs are $\chi$-bounded, some are not. It is not known whether there exists a hereditary class of graph that is $\chi$-bounded but not polynomially $\chi$-bounded. The goal of this talk is to survey several results, proof techniques, and open questions around the notion of polynomial $\chi$-boundedness.