The last few years have seen dramatic advances in the growth, fabrication and characterization of low-dimensional materials (such as graphene) and nanostructures (such as those formed from ultrathin films, wires, discs and other “dots”), formed either singly or in spatially periodic arrays. Most studies of these artificially engineered materials have been driven by their potential for device applications that involve smaller and smaller physical dimensions. In particular, the dynamical properties of these materials are of fundamental interest for the devices that involve high-frequency operation and/or switching. Consequently, the different excitations, vibrational, magnetic, optical, electronic, and so on, need to be understood from the perspective of how their properties are modified in finite structures especially on the nanometre length scale due to the presence of surfaces and interfaces. Recently, the patterning of nanoelements, into periodic and other arrays, has become a focus of intense activity, leading for example to photonic crystals and their analogues such as phononic and magnonic crystals where the control of the band gaps in the excitation spectrum is a basis for applications. The nonlinear properties of the excitations are increasingly a topic of interest, as well as the linear dynamics.