  Q-balls are regular extended `objects' that exist for some non-gravitating,
self-interacting, scalar field theories with a global, continuous, internal
symmetry, on Minkowski spacetime. Here, analogous objects are also shown to
exist around rotating (Kerr) black holes, as non-linear bound states of a test
scalar field. We dub such configurations Q-clouds. We focus on a complex
massive scalar field with quartic plus hexic self-interactions. Without the
self-interactions, linear clouds have been shown to exist, in synchronous
rotation with the black hole horizon, along 1-dimensional subspaces - existence
lines - of the Kerr 2-dimensional parameter space. They are zero modes of the
superradiant instability. Non-linear Q-clouds, on the other hand, are also in
synchronous rotation with the black hole horizon; but they exist on a
2-dimensional subspace, delimited by a minimal horizon angular velocity and by
an appropriate existence line, wherein the non-linear terms become irrelevant
and the Q-cloud reduces to a linear cloud. Thus, Q-clouds provide an example of
scalar bound states around Kerr black holes which, generically, are not zero
modes of the superradiant instability. We describe some physical properties of
Q-clouds, whose backreaction leads to a new family of hairy black holes,
continuously connected to the Kerr family.
