The dynamic self-assembly mechanism of fullerene molecules is an irreversible process emerging naturally under the nonequilibrium conditions of hot carbon vapor and is a consequence of the interplay between the dynamics and chemistry of polyyne chains, pi-conjugation and corresponding stabilization, and the dynamics of hot giant fullerene cages. In this feature article we briefly present an overview of experimental findings and past attempts to explain fullerene formation and show in detail how our recent quantum chemical molecular dynamics simulations of the dynamics of carbon vapor far from thermodynamic equilibrium have assisted in the discovery of the combined size-up/size-down "shrinking hot giant" road that leads to the formation of buckminsterfullerene C-60, C-70, and larger fullerenes. This formation mechanism is the first reported case of order created out of chaos where a distinct covalent bond network of an entire molecule is spontaneously self-assembled to a highly symmetric structure and fully explains the fullerene formation process consistently with all available experimental observations a priori. Experimental evidence suggests that it applies universally to all fullerene formation processes irrespective of the carbon source.