Conventional approaches to control and shape the scattering patterns of light generated by different nanostructures are mostly based on engineering of their electric response due to the fact that most metallic nanostructures support electric resonances in the optical frequency range. Recently, fuelled by the fast development in the fields of metamaterials and plasmonics, artificial optically-induced magnetic responses have been demonstrated for various nanostructures. This kind of response can be employed to provide an extra degree of freedom for the efficient control and shaping of the scattering patterns of nanoparticles and nanoantennas. Here we review the recent progress in this research direction of nanoparticle scattering shaping and control through the interference of both electric and optically-induced magnetic responses. We discuss the magnetic resonances supported by various structures in different spectral regimes, and then summarize the original results on the scattering shaping involving both electric and magnetic responses, based on the interference of both spectrally separated (with different resonant wavelengths) and overlapped dipoles (with the same resonant wavelength), and also other higher-order modes. Finally, we discuss the scattering control utilizing Fano resonances associated with the magnetic responses. ; Comment: To appear in Chinese Physics B (January, 2014), as an invited review paper for the topical review column titled "Plasmonics and Metamaterials"