The presentation surveys the results on the studies of the nonlinear-optical effects in different types of novel planar metamaterials utilizing Mie-type dielectric resonances. First, we discuss the observation enhanced third-harmonic generation from silicon nanodisks exhibiting both electric and magnetic dipolar resonances [1]. Experimental characterization of the nonlinear optical response through third-harmonic microscopy and spectroscopy reveals that the third-harmonic generation is significantly enhanced in the vicinity of the magnetic dipole resonances. Then, we have studied third-harmonic generation from lowloss subwavelength silicon nanodisks arranged in the form of trimer oligomers with varying distance between the nanoparticles. Each of the nanodisks exhibits both electric and magnetic Mie-type resonances that are shown to affect significantly the nonlinear response. We observe the third- harmonic radiation intensity that is comparable to that of a bulk silicon slab and demonstrate a pronounced reshaping of the third-harmonic spectra due to interference of the nonlinearly generated waves augmented by an interplay between the electric and the magnetic dipolar resonances [2].   The ultrafast all-optical switching in subwavelength nonlinear dielectric nanostructures exhibiting localized magnetic Mie resonances is also detected [3]. We employ amorphous silicon nanodisks to achieve strong self-modulation of femtosecond pulses with a depth of 60% at picojoule-per-disk pump energies. In the pump-probe measurements, we reveal that switching in the nanodisks can be governed by pulse-limited 65 fs-long two-photon absorption being enhanced by a factor of 80 with respect to the unstructured silicon film.   We have also analyzed the nonlinear scattering from dielectric clusters composed of four identical silicon nanodisks excited in the spectral range close to the magnetic dipole resonance of the individual disk [4]. We showed, for the first time to our knowledge, that resonant excitation of collective modes in such symmetric quadrumers results in substantial enhancement of the nonlinear response, as compared to that from both optically decoupled silicon disks and an unpatterned silicon film of the same thickness.