Metals and semiconductors are widely used for making technological devices as they carry electricity. Discovery of topological insulator (TI) has enriched the list of such materials . TIs are different than conventional insulators in the sense that, even though the bulk of the system remains insulating, the boundary of the system can carry current, due to the presence of conducting states. Moreover, these states are usually symmetry protected, and show robustness feature. This can be made use for making energy efficient spintronic devices, and fault-tolerant quantum computers. As the search for novel topological phases of matter is on the rise, this thesis mostly deals with the realization of non-interacting TI phases theoretically, in both real materials and toy model scenarios. In particular, it focuses on exploring the topological crystalline insulating (TCI) phases, which are basically TIs with protected crystalline symmetries. In connection to the TCI phases based on layered systems, chapter 2 discusses about how one can attain TCIs protected by mirror symmetry in heterostructures consisting of graphene monolayers separated by two-dimensional polar spacers. In chapter 3, we first focus on the naturally occurring mineral called Jacutingaite (Pt$_{2}$HgSe$_{3}$), and show based on density-functional calculations that it realizes dual topological phase (weak TI and TCI) and that the same conclusion holds for Pd$_{2}$HgSe$_{3}$. What makes this layered system more interesting is the fact that monolayer version of Jacutingaite is predicted to have a sizable band gap of $~0.5$eV, featuring a novel quantum spin Hall insulator. Further, we introduce tight-binding models that capture the essential topological properties of this dual topological phase in materials with three-fold rotation symmetry and use these models to describe the main features of the surface spectral density of different materials in the class. Following this, chapter 4 aims at exploring topological phases for two cubic three dimensional half-Heusler materials belonging to the space group 216. We investigate from first-principles the possibility of hinge modes in very proximate topological phases tunable by moderate uniaxial strain. We consider the compounds LiSbZn and LiBiZn. While LiSbZn has a topologically trivial band structure, the larger spin-orbit coupling of Bi causes a band inversion in LiBiZn. We predict the existence of topologically trivial hinge states in both cases. The hinge modes are affected by the bulk topological phase transitions, but in an indirect way: topological surface modes, when present, hybridize with the hinge states and obscure their visibility. Thus, we find that the most visible hinge modes actually occur when no band inversions are present in the material. Our work highlights the interplay and competition between surface and hinge modes in half-Heuslers, and may help guide the experimental search for robust boundary signatures in these materials. Finally, in chapter 5, the thesis presents a flavor of newly discovered topological transport phenomena, namely Magnus Hall effect in two and three dimensional systems. Effects of strain, warping, and tilt on response have been explored in detail. Starting with two-dimensional (2D) topological systems, we find that warping induced asymmetry in both the Fermi surface and Berry curvature can in general enhance the Magnus response for monolayer graphene and surface states of topological insulator. The strain alone is only responsible for Magnus valley responses in monolayer graphene while warping leads to finite Magnus response there. Interestingly, on the other hand, strain can change the Fermi surface character substantially that further results in distinct behavior of Magnus transport coefficients as we observe in bilayer graphene. And finally, going beyond 2D systems, we also investigate the Magnus responses in three-dimensional multi-Weyl semimetals (mWSMs) to probe the effect of tilt and anisotropic nonlinear energy dispersion. Remarkably, Magnus responses can only survive for the WSMs with chiral tilt. In particular, our study indicates that the chiral (achiral) tilt engenders Magnus (Magnus valley) responses. Therefore, Magnus responses can be used as a tool to distinguish between the untilted and tilted WSMs in experiments.