  We introduce a systematic and direct procedure to generate hairy rotating
black holes by deforming a spherically symmetric seed solution. We develop our
analysis in the context of the gravitational decoupling approach, without
resorting to the Newman-Janis algorithm. As examples of possible applications,
we investigate how the Kerr black hole solution is modified by a surrounding
fluid with conserved energy-momentum tensor. We find non-trivial extensions of
the Kerr and Kerr-Newman black holes with primary hair. We prove that a
rotating and charged black hole can have the same horizon as Kerr's,
Schwarzschild's or Reissner-Nordstr\"om's, thus showing possible observational
effects of matter around black holes.
