Delineates how the ideas of topological equivalence and adiabatic continuity lead to the emergence of distinct classes of insulator Hamiltonians, and how this, in turn, leads to bulk-boundary correspondence – the connection between bulk topological invariants and edge or surface states. Classification of topologically nontrivial and trivial phases, based on fundamental discrete symmetries and dimensionality, the “tenfold way," is explained. Mapping of d-dimensional Brillouin zones onto a d-dimensional Brillouin torus and Bloch Hamiltonians are defined. and construction of Bloch bundles on the torus base manifold is outlined. Time-reversal symmetry, Kramers’ band degeneracy, “time-reversal invariant momenta,” and the implied vanishing of Berry’s curvature are delineated. The integer quantum Hall effect and the modern theory of polarization are discussed in detail. Z2 topological invariant is derived using the sewing matrix, time-reversal polarization and the non-Abelian Berry connection.