A report is given of progress achieved during a 5-year period for describing the solidification of polymers by crystallization. Emphasis is placed on the fact that the classical quasi-equilibrium treatment of solidification under heat transfer conditions (as known under the heading of moving boundary value problems) is unable to predict any texturing in one-component systems. For polymers which are notorious for their slow approach to thermodynamic equilibrium, the local equilibrium theories, which are popular in metallurgy, are not discussed. In the present paper we show how to cope with nucleation under changing temperature conditions. Use is made of an adaptation of Avrami's theory to nonisothermal conditions, as recently proposed. Experimental work is extended to conditions of flow, where nucleation is enhanced. We show the possibility of theoretically describing this latter phenomenon.