In this paper, I describe the computer simulations that I have performed to critically examine the Lauritzen-Hoffman (LH) and the Sadler-Gilmer (SG) theories of polymer crystallization. In particular, I have computed the free energy profile for nucleation of a new crystalline layer on the growth face to compare with that assumed by the LH theory, I have analysed the mechanism of thickness selection in a multi-pathway model in which some of the constraints in the LH theory are relaxed, and I have re-examined the model used by SG. These investigations have lead to a mechanism of thickness selection of lamellar polymer crystals that differs from the two theories that I set out to examine.