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This paper is the eighth in a sequence on the structure of sets of solutions to systems of equations in free and hyperbolic groups, projections of such sets (Diophantine sets), and the structure of definable sets over free and hyperbolic groups. In the eighth paper we use a modification of the sieve procedure, that was presented in [Se6] as part of the quantifier elimination procedure, to prove that free and torsionfree (Gromov) hyperbolic groups are stable.doi:10.4007/annals.2013.177.3.1 fatcat:3nwvd4alpffshj27zz3nspsmma