Curve graphs are crucial tools for studying mapping class groups of surfaces. However, many basic questions on their geometry remain open. In this talk, we will shed light on the geometry of curve graphs by describing “filtrations” of them by hyperbolic graphs. These filtrations yield quasi-isometric disintegrations of curve graphs into trees. As a corollary, we provide a new proof of finite asymptotic dimension of curve graphs. Finally, we describe some useful aspects of the dynamics of the mapping class group actions on the graphs in the filtrations.