  In this paper we classify, up to orbit equivalence, cohomogeneity one actions
of connected closed Lie subgroups of $U(1,n)$ on the $(2n+1)$-dimensional anti
de Sitter spacetime $AdS^{2n+1}$. We also give some new examples of nonproper
cohomogeneity one actions on $AdS^{n+1}$ and determine parabolic Lie subgroups
of $SO(2,n)$ and their orbits in $AdS^{n+1}$
