DescriptionWe apply the quilted Floer theory of Wehrheim and Woodward to families of quilted surfaces parametrized by the Stasheff multiplihedra. Our approach is modeled on the construction of the Fukaya category, which applies Floer theory to families of pointed Riemann surfaces parametrized by the associahedra. First, we show that the multiplihedra are realized as a moduli space of quilted disks. Using the quilted disks we define the moduli space of pseudoholomorphic quilted disks, which under suitable transversality assumptions are smooth manifolds. Then we prove a gluing theorem relating ``broken'' tuples of pseudoholomorphic quilted disks with boundaries of one-parameter familes of pseudoholomorphic quilted disks.