Advances in material synthesis and in metamaterial technology offer new venues to tailor the electromagnetic properties of devices, which may go beyond conventional limits in a variety of fields and applications. Invisibility and cloaking are perhaps one of the most thought-provoking possibilities offered by these new classes of advanced materials. Here, recently proposed solutions for invisibility and cloaking using metamaterials, metasurfaces, graphene and/or plasmonic materials in different spectral ranges are reviewed and highlighted. The focus is primarily on scattering-cancellation approaches, describing material challenges, venues and opportunities for the plasmonic and the mantle cloaking techniques, applied to various frequency windows and devices. Analogies, potentials and relevant opportunities of these concepts are discussed, their potential realization and the underlying technology required to verify these phenomena are reviewed with an emphasis on the material aspects involved. Finally, these solutions are compared with other popular cloaking techniques.