The role of crack propagation velocity in the cleavage of crystals and aggregates is emphasized in this paper. The available data on crack propagation in elastic media are summarized, showing that this phenomenon is adequately understood at present. For inelastic media, it is postulated that in addition to the Griffith Criterion two critical velocity conditions must be satisfied or crack propagation cannot occur. One critical velocity pertains to propagation through a crystal, and the other to propagation through an aggregate. The first critical velocity depends on the rate at which plastic glide can absorb energy at a crack tip and the number and size of cleavage steps that a crack front contains. These quantities are estimated. The second critical velocity accounts for the dependence of fracture stress on grain size in aggregates and interprets other experimental behaviors of aggregates which contradict previous interpretations.