ABSTRACT Giant planets can clear deep gaps when embedded in 2D (razor-thin) viscous circumstellar disks. We show by direct simulation that giant planets are just as capable of carving out gaps in 3D. Surface density maps are similar between 2D and 3D, even in detail. In particular, the scaling of gap surface density with planet mass, derived from a global "zero-dimensional" balance of Lindblad and viscous torques, applies equally well to results obtained at higher dimensions. Our 3D simulations reveal extensive, near-sonic, meridional flows both inside and outside the gaps; these large-scale circulations might bear on disk compositional gradients, in dust or other chemical species. At high planet mass, gap edges are mildly Rayleigh unstable and intermittently shed streams of material into the gap-less so in 3D than in 2D.