In　this　article,　we　consider　the　wave　equation　on　hyperbolic　spaces　H-n(n　＞=　2)　with　nonlinear　locally　distributed　damping　as　follow:　{u(tt)　-　Delta(g)u　+　a(x)g(u(t))　=　0　(x,　t)　is　an　element　of　H-n　x　(0,+infinity),(1)　u(x,　0)　=　u(0)(x),　u(0)(x,　0)　=　u(1)(x)　x　is　an　element　of　H-n.　It　is　well-known　that　the　energy　of　the　system　(1)　is　of　polynomial　decay　in　the　Euclidean　space.　However,　on　hyperbolic　spaces,　owing　to　the　following　inequality　integral(Hn)　u(2)dx(g)　＜=　C　integral(Hn)　|del(g)u|(2)(g)dx(g),　f　or　u　is　an　element　of　H-1　(H-n),　(2)　we　prove　the　exponential　stabilization　of　the　wave　equation　by　multiplier　methods　and　compactness-uniqueness　arguments.