There exist well-known no-hair theorems forbidding the existence of hairy black hole solutions in general relativity coupled to a scalar conformal field theory in asymptotically flat space. Even in the presence of cosmological constant, where no-hair theorems can usually be circumvented and black holes with conformal scalar hair were shown to exist in D≤4 dimensions, no-go results were reported for D>4. In this paper we prove that these obstructions can be evaded and we answer in the affirmative a question that remained open: Whether hairy black holes do exist in general relativity sourced by a conformally coupled scalar field in arbitrary dimensions. We find the analytic black hole solution in arbitrary dimension D>4, which exhibits a backreacting scalar hair that is regular everywhere outside and on the horizon. The metric asymptotes to (anti-)de Sitter spacetime at large distance and admits spherical horizon as well as horizon of a different topology. We also find analytic solutions when higher-curvature corrections O(Rn) of arbitrary order n are included in the gravity action.