Standard models in physics are all based on particles.  All particles have a well-defined mass. This concept is so deeply engrained in our understanding of matter that it would be difficult to imagine a world in which stuff existed that, rather than having a well-defined mass, had instead all possible mass.  This paper will describe a series of materials in which such stuff, called unparticles, offers a natural explanation of their properties. However, the evidence for unparticles thus far is entirely circumstantial.  Should unparticles be found experimentally, this would radically alter our conception of the physical universe. A proposal for an experiment to detect unparticles directly will be presented.  The experiment turns on the fact that unparticles do not interact with light in the standard way and hence leave a fingerprint distinct from that of particle matter.