  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 dimensions three and four,
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(R^n) of arbitrary
order n are included in the gravity action.
