This paper introduces several cellular automata in hyperbolic spaces which are able to simulate any computational device, as they are universal. We sketchily present five universal cellular automata, four of them on two grids of the hyperbolic plane and one in a grid of the hyperbolic 3D space. Key–Words: Theory of Computation, Cellular Automata, universality, hyperbolic geometry.

Abstract A soft presentation of hyperbolic spaces (as metric spaces), free differential apparatus, is offered. Fifth Euclid’s postulate in such overthrown and, among other things, it proved that spheres (equipped with great-circle distances) and Euclidean are the only locally compact geodesic (i.e., convex) three-point homogeneous.