Ray Rappaport spent many years studying microtubule asters, and how they induce cleavage furrows. Here, we review recent progress on aster structure and dynamics in zygotes and early blastomeres of Xenopus laevis and Zebrafish, where cells are extremely large. Mitotic and interphase asters differ markedly in size, and only interphase asters span the cell. Growth of interphase asters occurs by a mechanism that allows microtubule density at the aster periphery to remain approximately constant as radius increases. We discuss models for aster growth, and favor a branching nucleation process. Neighboring asters that grow into each other interact to block further growth at the shared boundary. We compare the morphology of interaction zones formed between pairs of asters that grow out from the poles of the same mitotic spindle (sister asters) and between pairs not related by mitosis (non-sister asters) that meet following polyspermic fertilization. We argue growing asters recognize each other by interaction between antiparallel microtubules at the mutual boundary, and discuss models for molecular organization of interaction zones. Finally, we discuss models for how asters, and the centrosomes within them, are positioned by dynein-mediated pulling forces so as to generate stereotyped cleavage patterns. Studying these problems in extremely large cells is starting to reveal how general principles of cell organization scale with cell size.