  In this review, We discussed the theoretical foundation and experimental
discovery of different topological electronic states of material in condensed
matter. At first, we briefly reviewed the conventional electronic states, which
have been realized in band theory of solid. Next, the simplest non-trivial
insulating phase (Integer Quantum Hall State) and the concept of topological
order in condensed matter electronic system have been introduced. In the
following sections, we discussed Quantum Spin Hall (QSH) State in two
dimensions (2D), and reviewed the theoretical and experimental developments
from 2D QSH state to 3D topological insulators (TI). Subsequently, we gave a
brief overview on theoretical and experimental understanding on recently
discovered topological Dirac semimetals, Weyl semimetals, three-, six- , and
eight-fold degenerate semimetals, and Nodal line semimetals. Then, topological
crystalline insulator, which can not be considered as a descendent of Quantum
Spin Hall or Integer Quantum Hall insulator, has been introduced. Finally, we
discussed the presence of magnetism in some topological materials and its
consequence on electronic band structure.
