Inverse problems arise in all fields of science and technology, where information is obtained about an object from a set of observed measurements. Taking human vision as an example, our brains construct a detailed, 3-D map from measurements of scattered light that reach our retinas. Thus, solving inverse problems can reveal what is hidden. But can objects be made invisible? Fascination with this subject includes Greek mythology and science fiction like Star Trek and Harry Potter. This lecture will explore several inverse problems and then describe a simple and powerful proposal for achieving invisibility, the so-called transformation optics, and other progress that has been made in the last decade.