A graph $G$ is said to be $H$-free if it contains no induced subgraph isomorphic to $H$. A family of graphs $\mathcal{G}$ is said to be $\chi$-bounded if there exists some function $f$ such that $\chi(G)\leq f(\omega(G))$ for every $G\in\mathcal{G}$, and $f$ is said to be the binding function of $G$. In this talk, we will talk about the binding functions of graphs with some specific forbidden configurations, and we present some resent results on coloring $P_5$-free graphs.