Until recently it has been well understood that there are only two types of insulators. One kind is band insulator consistent with band theory of solid state physics, and the other is Mott insulator which is supposed to be metal based on the band theory, but it turns out to be insulating due to strong interaction between electrons. Very recently another “conceptually” distinct type of insulator was proposed named a topological insulator. A key character of this insulator is a conducting edge state while the bulk state has a band gap, and a main ingredient of being topological insulators is spin-orbit coupling. There are many materials suggested for candidates of topological insulators. There have been recent attempts to search for topological characters in metallic systems as well. See our papers on arXiv. org.