We characterize the response of isolated single-wall (SWNT) and multiwall (MWNT) carbon nanotubes and nanotube bundles to static electric fields using first-principles calculations and density-functional theory. The longitudinal polarizability of SWNTs scales as the inverse square of the band gap, while in MWNTs and bundles it is given by the sum of the polarizabilities of the constituent tubes. The transverse polarizability of SWNTs is insensitive to band gaps and chiralities and is proportional to the square of the effective radius; in MWNTs, the outer layers dominate the response. The transverse response is intermediate between metallic and insulating, and a simple electrostatic model based on a scale-invariance relation captures accurately the first-principles results. The dielectric response of nonchiral SWNTs in both directions remains linear up to very high values of applied field.