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Periodical cicadas present numerous puzzles for biologists. First, their period is fixed, with individuals emerging as adults precisely after either 13 or 17 years (depending on species). Second, even when there are multiple species of either 13- or 17-year cicadas at the same location, only one or rarely two broods (cohorts) co-occur, so that periodical cicada adults appear episodically. Third, the 13- or 17-year periods of cicadas suggest there is something important about prime numbers. Finally, single broods can dominate large areas, with geographical boundaries of broods remaining generally stable through time. While previous mathematical models have been used to investigate some of these puzzles individually, here we investigate them all simultaneously. Unlike previous models, we take an explicitly evolutionary approach. Although not enough information is known about periodical cicadas to draw firm conclusions, the theoretical arguments favor a combination of predator satiation and nymph competition as being key to the evolution of strictly fixed periods and occurrence of only one brood at most geographical locations. Despite ecological mechanisms that can select for strictly fixed periods, there seem to be no plausible ecological mechanisms that select for periods being prime numbers. This suggests that the explanation for prime-numbered periods, rather than just fixed periods, may reside in physiological or genetic mechanisms or constraints. Key words: Allee effects; evolution of periodicity; Magicicada; rock--paper--scissors competition; spatial dynamics.