A very big story in recent years in condensed matter physics is that of topological insulators, and it's a great tale of finding something new "in plain sight". For something like 70 or 80 years, physicists thought that they had a handle on the insulating state. Take a large crystalline solid, and ignore electron-electron interactions for the moment. The allowed electronic states for such a material come in bands, when you look at how they're distributed as a function of energy. That is, there are many electronic states clustered so close together in energy, separated by energy gaps where there are no allowed electronic states. Now consider filling up those states with some number of electrons, counting two electrons (one spin up, one spin down) per state. (This is short-hand for something more sophisticated, but it gets the point across, just as filling up atomic orbitals does a pretty good job describing the periodic table.) If you end up in the middle of a band, with lots of empty states right next to the filled states in energy, then you have a metal. If you end up exactly filling a band, then you have either a band insulator (if the energy gap next to the most energetic filled state is several eV) or a semiconductor (if the energy gap is more like 1-3 eV). Turning on electron-electron interactions can change things a bit, but not too much. (Interactions can lead to one more kind of insulator, a Mott insulator, in which interactions open up a gap in what would otherwise have been a metallic system.)
Until recently, we thought that this was all there was to it, as far as band insulators go. It turns out that this is not the case, because of what happens at the boundary of the material (which we have so far been ignoring). In some band insulators, the surface states (in 3d) or edge states (in 2d) can have special properties. For example, one could have a situation where (because of spin-orbit coupling + band structure) the right-moving charge carriers have to have their spin pointed in one direction, while the left-moving charge carriers have to have their spin pointed the opposite way. The result is that these surface states with unusual Dirac-like dispersion are thought to be able to resist back-scattering very effectively. This means that these surface states may be very interesting for electronics applications, having ballistic properties over long distances. Moreover, these properties are expected to be rather robust, because they come from the topology of the states, which is not easily disturbed by disorder. For great reviews of this, see this paper (soon to appear in RMP), this article in Physics Today, and this video.
There is evidence that these states do exist, particularly from surface scattering techniques such as ARPES. Transport experiments have faced a challenge, however, since many of the candidate materials (Bi2Se3, for example) are difficult to grow in sufficient purity that the bulk is actually insulating. Still, this is exciting stuff, and a new paper on the arxiv (1003.0155) reveals that there may be a whole new class of other materials to play with. Surprises from nature are always fun.