For centuries, the conventional approach to lens design has been to grind the surfaces of a uniform material in such a manner as to sculpt the paths that rays of light follow as they transit through the interfaces. Refractive lenses formed by this procedure of bending the surfaces can be of extremely high quality, but are nevertheless limited by geometrical and wave aberrations that are inherent to the manner in which light refracts at the interface between two materials. Conceptually, a more natural-but usually less convenient-approach to lens design would be to vary the refractive index throughout an entire volume of space. In this manner, far greater control can be achieved over the ray trajectories. Here, we demonstrate how powerful emerging techniques in the field of transformation optics can be used to harness the flexibility of gradient index materials for imaging applications. In particular we design and experimentally demonstrate a lens that is broadband (more than a full decade bandwidth), has a field-of-view approaching 180, and zero f-number. Measurements on a metamaterial implementation of the lens illustrate the practicality of transformation optics to achieve a new class of optical devices.