Originally regarded as forbidden, black hole "hair" are fields associated with a stationary black hole apart from the gravitational and electromagnetic ones. Several stable stationary black hole solutions with gauge or Skyrme field hair are known today within general relativity. We formulate here a "no scalar-- hair" conjecture, and adduce some new theorems that almost establish it by ruling out - for all but a small parameter range - scalar field hair of spherical black holes in general relativity, whether the field be self--interacting, coupled to an Abelian gauge field, or nonminimally coupled to gravity. Twenty--five years ago Wheeler enunciated the Israel--Carter conjecture, today colloquially known as "black holes have no hair" . Inmensely influential in black hole physics, this conjecture has long been regarded as a theorem by large sectors of the gravity--particle physics community. On the other hand, the proliferation in the 1990's of solutions for stationary black ..