We describe recent theoretical and experimental progress on making objects invisible. The equations that govern a variety of wave phenomena, including electrostatics, electromagnetism, acoustics and quantum mechanics, have transformation laws that allow one to design material parameters that steer waves around a hidden region, returning them to their original path on the far side. Not only are observers unaware of the contents of the hidden region, they are not even aware that something was being hidden; the object, which casts no shadow, is said to be cloaked. We recount the recent history of the subject and discuss some of the mathematical issues involved.