  Scalar and gravitational perturbations on Kerr-anti-de Sitter (Kerr-AdS)
black holes have been addressed in the literature and have been shown to
exhibit a rich phenomenology. In this paper we complete the analysis of bosonic
fields on this background by studying Maxwell perturbations, focusing on
superradiant instabilities and vector clouds. For this purpose, we solve the
Teukolsky equations numerically, imposing the boundary conditions we have
proposed in\cite{Wang:2015goa} for the radial Teukolsky equation. As found
therein, two Robin boundary conditions can be imposed for Maxwell fields on
Kerr-AdS black holes, one of which produces a new set of quasinormal modes even
for Schwarzschild-AdS black holes. Here, we show these different boundary
conditions produce two different sets of superradiant modes. Interestingly the
"new modes" may be unstable in a larger parameter space. We then study
stationary Maxwell clouds, that exist at the threshold of the superradiant
instability, with the two Robin boundary conditions. These clouds, obtained at
the linear level, indicate the existence of a new family of black hole
solutions at the nonlinear level, within the Einstein-Maxwell-AdS system,
branching off from the Kerr-Newman-AdS family. As a comparison with the Maxwell
clouds, scalar clouds on Kerr-AdS black holes are also studied, and it is shown
there are Kerr-AdS black holes that are stable against scalar, but not vector
modes, with the same "quantum numbers".
