Here, we introduce the concept of magnetic localized surface plasmons (LSPs), magnetic dipole modes that are supported by cylindrical metal structures corrugated by very long, curved grooves. The resonance wavelength is dictated by the length of the grooves, allowing us to tune it to values much larger than the size of the particle. Moreover, magnetic LSPs also exist for extremely thin metal disks and, therefore, they could be used to devise metasurfaces with magnetic functionalities. Experimental evidence of the existence of these magnetic LSPs in the microwave regime is also presented, although the concept is very general and could be applied to terahertz or infrared frequencies.