We study spacelike entire constant mean curvature hypersurfaces in Anti-de Sitter space of any dimension. First, we give a classification result with respect to their asymptotic boundary, namely we show that every admissible sphere $\Lambda$ is the boundary of a unique such hypersurface, for any given value $H$ of the mean curvature. We also demonstrate that, as $H$ varies in $\mathbb{R}$, these hypersurfaces analytically foliate the invisible domain of $\Lambda$. Finally, we extend Cheng-Yau Theorem to the Anti-de Sitter space, which establishes the completeness of any entire constant mean curvature hypersurface.