  We construct and study rotating axion boson stars (RABSs). These are the
spinning generalisations of the spherical gravitating solitons recently
introduced in arXiv:1909.05515. RABSs are asymptotically flat, stationary,
axially symmetric, everywhere regular solutions of the Einstein-Klein-Gordon
theory, in the presence of a periodic scalar potential arising in models of
axion-like particles. The potential is characterised by two parameters: the
mass of the scalar field $m_a$ and the decay constant $f_a$. We present an
overview of the solution space, for different values of $f_a$, and analyse some
of their phenomenological properties. For large decay constants the solutions
become identical to the standard spinning mini boson stars. For small decay
constants, on the other hand, the solutions develop distinctive features. In
particular, we analyse their compactness, the emergence of ergoregions, light
rings and the distribution of stable and unstable equatorial timelike circular
orbits, including the innermost stable circular orbit. We also establish the
occurrence of violations of the strong energy condition for physical observers,
for some RABSs. We observe some analogy between RABSs and spinning gravitating
$Q$-balls.
