We present a survey of results from various research groups under the unifying viewpoint of transformational physics, which has been recently introduced for the design of metamaterials in optics and acoustics. We illustrate the versatility of underlying geometric transforms in order to bridge wave phenomena (the different 'colours' of waves) ranging from transverse electric waves, to linear surface water waves at an air-fluid interface, to pressure waves in fluids and out-of-plane shear waves in elastic media: these waves are all governed by a second order scalar partial differential equation (PDE) invariant under geometric transform. Moreover, flexural waves propagating in thin plates represent a very peculiar situation whereby the displacement field satisfies a fourth order scalar PDE which also retains its form under geometric transform (unlike for the Navier equation in elastodynamics). Control of flexural wave trajectories is illustrated with a multilayered cloak and a carpet. Interestingly, the colours of waves can be revealed through an analysis of the band spectra of invisibility cloaks. In the context of acoustics, this suggests one can hear the shape of a drum. Alternative avenues towards cloaking based upon anomalous resonances of a negatively refracting coating (which can be seen as the result of folding the space back onto itself), and even plasmonic shells reducing the scattering cross-section of nano-objects are also addressed.