Recently, Algom-Kfir and Bestvina introduced mapping class groups of locally finite graphs as a proposed analog of Out(Fn) in the infinite-type setting. In this talk we will introduce the classification of infinite-type graphs, their mapping class groups, and some important types of elements in these groups. Using a framework established by Rosendal, we will then classify the coarse geometry of the pure mapping class groups. This is joint work with George Domat and Sanghoon Kwak.