Shape transformations and topological changes of lipid vesicles, such as fusion, budding, and fission, have important chemical physical and biological significance. In this paper, we study the fission process of lipid vesicles. Two distinct routes are considered that are both based on an asymmetry of the lipid distribution within the membrane. This asymmetry consists of a nonuniform distribution of two types of lipids. In the first mechanism, the two types of lipids are equally distributed over both leaflets of the membrane. Phase separation of the lipids within both leaflets, however, results in the formation of rafts, which form buds that can split off. In the second mechanism, the asymmetry consists of a difference in composition between the two monolayers of the membrane. This difference in composition yields a spontaneous curvature, reshaping the vesicle into a dumbbell such that it can split. Both pathways are studied with molecular dynamics simulations using a coarse-grained lipid model. For each of the pathways, the conditions required to obtain complete fission are investigated, and it is shown that for the second pathway, much smaller differences between the lipids are needed to obtain fission than for the first pathway. Furthermore, the lipid composition of the resulting split vesicles is shown to be completely different for both pathways, and essential differences between the fission pathway and the pathway of the inverse process, i.e., fusion, are shown to exist.