For nearly 30 years, fast calcium waves have been attributed to a regenerative process propagated by CICR (calcium-induced calcium release) from the endoplasmic reticulum. Here, I propose a model containing a new subclass of fast calcium waves which is propagated by CICI (calcium-induced calcium influx) through the plasma membrane. They are called fast CICI waves. These move at the order of 100 to 1000 microm/s (at 20 degrees C), rather than the order of 3 to 30 microm/s found for CICR. Moreover, in this proposed subclass, the calcium influx which drives calcium waves is relayed by stretch-activated calcium channels. This model is based upon reports from approx. 60 various systems. In seven of these reports, calcium waves were imaged, and, in five of these, evidence was presented that these waves were regenerated by CICI. Much of this model involves waves that move along functioning flagella and cilia. In these systems, waves of local calcium influx are thought to cause waves of local contraction by inducing the sliding of dynein or of kinesin past tubulin microtubules. Other cells which are reported to exhibit waves, which move at speeds in the fast CICI range, include ones from a dozen protozoa, three polychaete worms, three molluscs, a bryozoan, two sea urchins, one arthropod, four insects, Amphioxus, frogs, two fish and a vascular plant (Equisetum), together with numerous healthy, as well as cancerous, mammalian cells, including ones from human. In two of these systems, very gentle local mechanical stimulation is reported to initiate waves. In these non-flagellar systems, the calcium influxes are thought to speed the sliding of actinomyosin filaments past each other. Finally, I propose that this mechanochemical model could be tested by seeing if gentle mechanical stimulation induces waves in more of these systems and, more importantly, by imaging the predicted calcium waves in more of them.