The past 10 years have seen remarkable developments in microwave, optical and acoustic methods that make use of the unique wave transmission properties of a class of ordered composites known as metamaterials. These materials differ from regular materials in that wave propagation is dramatically affected by the size and arrangement of their small-scale structure, and not just by the choice of constituent materials. Their characteristic feature sizes can be much less than, or on the same order as a wavelength. They have a variety of potential applications ranging from super-lenses, to enabling subwavelength imaging resolution, to the creation of a Harry Potter-like ‘invisibility cloak’. In this review we provide an overview of some of these achievements, much of which has been proposed and demonstrated with optical and electromagnetic waves. However, more recently these ideas have been extended to acoustic waves, offering the opportunity of controlling ultrasound wave propagation in a unique manner.