Magnetic linear response properties for achiral and chiral carbon nanotubes were investigated with first-principles electronic structure methods. We have computed the magnetic shielding inside and outside the tubes originating from electronic current densities induced by the application of an external magnetic field. This electronic response of the nanotubes was analyzed for external magnetic fields both parallel and perpendicular to the tube axis. The magnetic screening would be experienced by guest molecules inside the tubes, measurable by NMR spectroscopy on isotopically labeled samples. Special attention is given to chiral nanotubes, in which longitudinal fields induce a non-zero longitudinal current density and thus tangential magnetic fields outside the tubes.