Counterintuitive to a naive understanding of Darwinian evolution, where among two interacting species one is expected to be fitter than the other and therefore outcompetes it, a surprising biodiversity exists within the earth's ecosystems. Rock-paper-scissors games, where three strategies cyclically dominate each other, have emerged as a fruitful metaphor for the explanation of biodiversity. In this talk we discuss populations spatially coevolving with local cyclic dominance, and show that they are capable of preserving coexistence of all subpopulations, and in this way ensuring biodiversity. We find that the individuals' mobility competes with the locality of interactions (cyclic dominance) such that biodiversity gets lost above a certain mobility threshold. Below this critical value, all subpopulations coexist forming fascinating moving patterns composed of entangled spirals, which we describe analytically.