  Topological matter has become one of the most important subjects in
contemporary condensed matter physics. Here, I would like to provide a
pedagogical review explaining some of the main ideas, which were pivotal in
establishing topological matter as such an important subject. Specifically, I
explain how the integer quantum Hall state played the role as a prototype for
topological insulator, eventually leading to the concept of topological matter
in general. The topological nature of the integer quantum Hall state is best
represented by the Thouless-Kohmoto-Nightingale-den Nijs, or so-called TKNN
formula, which connects between the Berry phase and the Hall conductivity. The
topological non-triviality of topological insulator stems from the existence of
a Dirac monopole in an appropriate, but often hidden Hamiltonian parameter
space. Interestingly, having the identical Dirac monopole structure, the
Hamiltonian describing the Rabi oscillation bears the essence of topological
insulator. The concept of topological matter has expanded to include
topological semimetals such as Weyl and Dirac semimetals. A final frontier in
the research of topological matter is the interaction-induced topological
phases of matter, namely, the fractional Chern and topological insulators. The
existence of the fractional Chern and topological insulators has been proposed
theoretically by drawing an analogy from the fractional quantum Hall states.
The gist of this proposal is explained along with some of its issues. I
conclude this review by discussing some of the future directions in the
research of topological matter.

Strongly Correlated Electrons