The AIM SQuaRE “The global dynamics of Thurston’s pullback map” with participants, William Floyd, Gregory Kelsey, Sarah Koch, Russell Lodge, Walter Parry, and Kevin Pilgrim, sought to classify the particular dynamical systems known as nearly Euclidean maps. The late mathematician William Thurston studied—among many other things—dynamical systems resulting from repeated iteration of a mapping from the two-dimensional sphere to itself. Applying a mapping over and over again, and watching what happens to a point, can produce fantastic and beautiful patterns that are intriguing to mathematicians. He focused on some particularly nice ones, the Euclidean Thurston maps, which are fully understood and classified using the tools and concepts of linear algebra. But much less well-understood is the much larger family of nearly Euclidean Thurston (NET) maps—the very simplest generalizations of the Euclidean ones. These are the dynamical systems studied by this SQuaRE.