Progression through the eukaryotic cell cycle is controlled at a series of checkpoints guarding transitions from one phase of the cycle to the next, e.g., G1-to-S, G2-to-M, metaphase-to-anaphase. These checkpoints ensure that a cell has satisfied certain requirements that are necessary for success of the next phase, e.g., that any DNA damage is repaired before the cell replicates its chromosomes in S phase. These transitions are irreversible: as soon as the conditions of the checkpoint are satisfied, the cell proceeds to the next phase and does not subsequently back up to the immediately preceding phase. The irreversibility of these transitions gives the cell cycle its directionality (G1 → S → G2 → M → G1 ...). The genes and proteins governing these checkpoints have been discovered by molecular geneticists, but the mechanistic basis of irreversibility is still a subject of controversy. Many molecular biologists think that the transitions are irreversible because key proteins are chemically degraded at each transition, but we maintain that irreversibility is a consequence of bistability and hysteresis in the underlying regulatory network. To prove this claim, JJT will describe the mechanism of the G1-S transition in some detail, build and analyze a mathematical model of the mechanism, and compare the implications of the model to experimental facts.