The majority of work on asymptotically anti-de Sitter
spacetimes, much of it motivated by the AdS/CFT correspondence, assumes
configurations which are either at or close to equilibrium.
Recently it has been discovered that genuinely dynamical
spacetimes with AdS asymptotics can exhibit some surprising behaviour. In
particular, numerical studies suggest AdS is dynamically unstable. I will
discuss recent analytical (in)stability results for asymptotically AdS
spacetimes, in particular how stability can be affected by boundary conditions
at the timelike conformal infinity.