We consider moduli spaces of quilted strips with markings and their compactifications. Using the theory of moment maps of toric varieties we identify the compactified moduli spaces with certain graph associahedra. We demonstrate how these moduli spac
es govern the combinatorics of A-infinity n-modules, which are natural generalizations of A-infinity modules (n=1) and bimodules (n=2).
We construct families of quilted surfaces parametrized by the multiplihedra, and define moduli spaces of pseudoholomorphic quilted disks using the theory of pseudoholomorphic quilts of Wehrheim and Woodward. We prove a gluing theorem for regular, iso
lated pseudoholomorphic quilted disks. This analytical result is a fundamental ingredient for the construction of A-infinity functors associated to Lagrangian correspondences.