Superfluidity - the ability of {\it liquid\/} $^4$He, when cooled
below 2.176~K, to flow without resistance through narrow pores - is one 
of the most amazing phenomena in physics.  Supersolidity - the coexistence 
of superfluid behavior with the crystalline order of a solid---
was proposed theoretically long ago as an even more exotic phase of 
{\it solid\/} $^4$He, but it has eluded detection until recently.

In 2004, Kim and Chan E. Kim and M. H. W. Chan [Nature (London) {\bf 427}, 225 (2004)] 
and E. Kim and M. H. W. Chan [Science {\bf 305}, 1941 (2004)] reported the 
onset of "nonclassical rotational inertia" in a torsional oscillator 
experiment with solid $^4$He, and they interpret their results as 
indicating the onset of supersolidity.

In this talk, I'll describe what a supersolid is, discuss the Chan et al
experiments (in the process revealing how to tell a raw from a hard boiled
egg), and present the theory I've recently developed (with Paul Goldbart of
University of Illinois at Urbana-Champaign, and Alan Dorsey of University of
Florida) of the normal solid to supersolid (NS-SS) phase transition.