We study the Mie scattering by high-index dielectric cylinders and reveal a cloaking mechanism associated with the resonant scattering spectra. The cloaking effect at any angle of observation originates from interference between a broadband radiation and narrow resonant Mie modes resulting in the Fano resonances. From the Fano formula it is clear that at the special frequency the numerator becomes zero and the scattering efficiency completely vanishes. The numerical calculations show that for a particular frequency the resonant Hz field near the boundary inside the high-index dielectric cylinders ideally mimics the incident field distribution and one observes a practically complete suppression of scattering. We verify the concept of Fano cloaking experimentally by monitoring the scattering by a water cylinder from visible to invisible regimes when its dielectric permittivity is changed by heating.