  We construct and analyse Kerr black holes (BHs) with synchronised axionic
hair. These are the BH generalisations of the recently constructed rotating
axion boson stars arXiv:2005.05982. Such BHs are stationary, axially symmetric,
asymptotically flat solutions of the complex Einstein-Klein-Gordon theory with
a QCD axion-like potential. They are regular everywhere on and outside the
event horizon. The potential is characterised by two parameters: the mass of
the axion-like particle, $m_a$ and the decay constant $f_a$. The limit $f_a
\rightarrow \infty$ recovers the original example of Kerr BHs with synchronised
scalar hair arXiv:1403.2757. The effects of the non-linearities in the
potential become important for $f_a \lesssim 1$. We present an overview of the
parameter space of the solutions together with a study of their basic geometric
and phenomenological properties, for an illustrative value of the coupling that
yields a non-negligible impact of the self-interactions.
