The aim of this paper is to develop a theory for the growth of multiple crystals in a polymer melt. This leads to nonlinear moving boundary problems for the temperature, with normal growth speed of the crystal boundaries determined by a nonlinear Gibbs-Thompson relation. Particular attention is paid to the effect of impingement, i.e., the event of two crystals hitting each other, which stops the growth on the contact interface. In one space dimension, the well-posedness of the growth model coupled to the heat equation is shown for an arbitrary number of crystals, both in a quasi-stationary and in an instationary situation. The resulting evolution of a fixed crystal in presence of other crystals is compared to the pure single crystal case. Finally, some basic features of the model in higher spatial dimensions and the main problems encountered in the attempt to prove a general well-posedness result are discussed.