Spherulitic growth and the microstructure of crystalline polymers is proved to correspond to space filling by the expansion of spheres. Therefore spherulitic microstructures are analogous to space filled in by randomly nucleating (in space), expanding and touching spheres. This is contrasted with microstructures in metals, where surface tension leads to reshaping of the grain boundaries. Geometrical solutions are given for the general cases of grain-boundary surface shapes depending on the ratio of growth rates and contact radii. A space-filling computer model for expanding spheres is described, and the program＇s URL on the Internet is given. Several micrographs and figures an analysed to show the relationship between grain-boundary shapes and growth kinetics. Previous papers and computer models on this subject are reviewed.