  Symmetry fractionalization describes the fascinating phenomena that
excitations in a 2D topological system can transform under symmetry in a
fractional way. For example in fractional quantum Hall systems, excitations can
carry fractional charges while the electrons making up the system have charge
one. An important question is to understand what symmetry fractionalization
(SF) patterns are possible given different types of topological order and
different symmetries. A lot of progress has been made recently in classifying
the SF patterns, providing deep insight into the strongly correlated
experimental signatures of systems like spin liquids and topological
insulators. We review recent developments on this topic. First, it was shown
that the SF patterns need to satisfy some simple consistency conditions. More
interesting, it was realized that some seemingly consistent SF patterns are
actually `anomalous', i.e. they cannot be realized in strictly 2D systems. We
review various methods that have been developed to detect such anomalies.
Applying such an understanding to 2D spin liquid allows one to enumerate all
potentially realizable SF patterns and propose numerical and experimental
probing methods to distinguish them. On the other hand, the anomalous SF
patterns were shown to exist on the surface of 3D systems and reflect the
nontrivial order in the 3D bulk. We review examples of this kind where the bulk
states are topological insulators, topological superconductors, or have other
symmetry protected topological orders.
