  In primary school, we were told that there are four states of matter: solid,
liquid, gas, and plasma. In college, we learned that there are much more than
four states of matter. For example, there are ferromagnetic states as revealed
by the phenomenon of magnetization and superfluid states as defined by the
phenomenon of zero-viscosity. The various phases in our colorful world are so
rich that it is amazing that they can be understood systematically by the
symmetry breaking theory of Landau. In this paper, we will review the progress
in last 20 -- 30 years, during which we discovered that there are even more
interesting phases that are beyond Landau symmetry breaking theory. We discuss
new "topological" phenomena, such as topological degeneracy, that reveal the
existence of those new phases - topologically ordered phases. Just like
zero-viscosity defines the superfluid order, the new "topological" phenomena
define the topological order at macroscopic level. As a new type of order,
topological order requires a new mathematical frame work, such as fusion
category and group cohomology, to describe it. More recently, we find that, at
microscopical level, topological order is due to long-range quantum
entanglements, just like fermion superfluid is due to fermion-pair
condensation. Long-range quantum entanglements lead to many amazing emergent
phenomena, such as fractional quantum numbers, fractional/non-Abelian
statistics, and perfect conducting boundary channels. Long-range quantum
entanglements can even provide a unified origin of light and electrons (or more
generally, gauge interactions and Fermi statistics): light waves (gauge fields)
are fluctuations of long-range entanglements, and electrons (fermions) are
defects of long-range entanglements.
