A marked surface is a compact surface, possibly with boundary, with a finite set of distinguished points called marked points. Triangulations of a marked surface, having the marked points as vertices, serve as a model for an associated cluster algebra. On the other hand, "noncrossing partitions of a cycle", which model the interval [1,c] in the absolute order on the symmetric group, can be viewed as noncrossing partitions of disk with marked points on its boundary. Joint work with Brestensky modeled the interval [1,c] in the affine symmetric group by noncrossing partitions of an annulus. In fact, the most natural notion of noncrossing partitions of an annulus models a lattice strictly containing [1,c] that was constructed by McCammond and Sulway to prove long-conjectured properties of Euclidean Artin groups. In this talk, I will discuss noncrossing partitions of marked surfaces, which generalize noncrossing partitions of the disk and the annulus. Partially ordered in the natural way, these form a lattice when there are no marked points in the interior of the surface. I will also discuss symmetric noncrossing partitions of a surface with double points, which model the interval [1,c] in the other classical affine Coxeter groups (and almost model McCammond and Sulway's lattice). Time allowing, I will mention work in progress with Eric Hanson to realize McCammond and Sulway's lattice in the representation theory of hereditary algebras of affine type.