The study of electronic properties of materials is of great interest to condensed matter physicists. Since the discovery of quantum Hall effects, band topology has attracted much attention. In this dissertation, the roles of band topology in the electronic and optical properties of layered materials and their moir´e structures are studied. In the first part of this dissertation, we study topological superconductivity in two dimensions, where Majorana bound states can be found at the corners of a material. In view of the bulk-edge correspondence, we propose the first theoretical model and show that a squareshaped 2D topological insulator proximitized by an s±-wave (e.g., Fe-based) superconductor can host a Majorana Kramers pair at each corner. Furthermore, the effects of the orientation of the edge and symmetry relaxations are addressed in a phase diagram. We also propose two possible experimental realizations of the corner states, providing a new avenue to explore non-Abelian quasiparticles. The second part focuses on twisted bilayer graphene (TBG), a system formed by stacking two sheets of graphene with a small twist angle. In TBG, the resulting moir´e pattern leads to a long-period superlattice structure. The theoretical prediction of flat bands under the low energy continuum model and the subsequent experimental discovery of strongly correlated phenomena near the magic angle (∼ 1.1 ◦ ) have sparked tremendous interests in this unprecedented system. Using the continuum model that captures the characteristics of TBG around the Dirac points known as K and K valleys, we first study the system at small twist angle ∼ 0.93◦ , which in experiment exhibits superconductivity and the correlated insulating behavior. At larger angles, (1.1 ◦ ∼ 2 ◦ ), we discovered two nontrivial Z2 topological invariants and show that one ensures the Dirac cones and the other leads to a non-local dissipative transport channel, which has been demonstrated by transport experiments in a series of devices with different twist angles. Our findings not only provide a new perspective for understanding the strongly correlated behavior in TBG but also suggest a potential strategy to achieve topological metamaterials from van der Waals materials. Moreover, we show that the enhanced density of states of moir´e bands, together with the presence of superlattice gaps, leads to a usually strong optical absorbance. Consquently, maximum extrinsic photoresponsivity of 26mAW−1 at 12 µm has been achieved in experiment, when the Fermi level is tuned to the superlattice band gap in 1.81◦ TBG, demonstrating the promising optical properties of TBG, and providing an alternative platform for tunable mid-infrared optoelectronics. All introductions, conclusions, and discussions in this dissertation do not reflect the viewpoints of any member of my supervisory committee.