  We present a family of solutions of Einstein's gravity minimally coupled to a
complex, massive scalar field, describing asymptotically flat, spinning black
holes with scalar hair and a regular horizon. These hairy black holes (HBHs)
are supported by rotation and have no static limit. Besides mass M and angular
momentum J, they carry a conserved, continuous Noether charge Q measuring the
scalar hair. HBHs branch off from the Kerr metric at the threshold of the
superradiant instability and reduce to spinning boson stars in the limit of
vanishing horizon area. They overlap with Kerr black holes for a set of (M,J)
values. A single Killing vector field preserves the solutions, tangent to the
null geodesic generators of the event horizon. HBHs can exhibit sharp physical
differences when compared to the Kerr solution, such as J/M^2>1, quadrupole
moment larger than J^2/M and larger orbital angular velocity at the innermost
stable circular orbit. Families of HBHs connected to the Kerr geometry should
exist in scalar (and other) models with more general self interactions.
