Over the past two decades, the discovery of topological Dirac materials, such as topological insulators and semimetals, has defined a new paradigm in condensed matter physics and established new platforms for realizing spin electronic devices and quantum computing. The novel quantum properties of these materials, which are mainly due to the presence of strong spin-orbit coupling, are invariant under topological transformations, making them more resilient to structural disorder, electromagnetic and thermal fluctuations in comparison to the conventional silicon-based materials. Their topological nature, characterized by topological invariants, depends crucially on the global symmetries of the system, with time-reversal symmetry and inversion symmetry being the most important ones. If these symmetries are broken by external fields, it is possible to cause transitions between different topological phases. One of the main challenges in this field is to determine the physical conditions under which different non-trivial topological phases can be realized, detected, and manipulated.

The present thesis aims at investigating the onset of different topological phases and associated quantum phenomena in topological material nanostructures such as thin films and nanoribbons where time-reversal symmetry is broken by the presence of magnetism. Specifically, using theoretical and computational methods based on atomistic tight-binding models and density functional theory, we have addressed four problems: (i) the possibility of realizing both the Chern insulator and the axion insulator phase in the same heterostructure consisting of a TI thin film sandwiched by two antiferromagnetic layers; (ii)  the origin of the deviations from exact quantization of the elusive topological magneto-electric effect in TI thin films in the axion insulator phase. For this purpose we have introduced a novel approach based on a non-local side-wall response that treats the quantum anomalous Hall effect and the topological magnetoelectric effect on the same footing; (iii) signatures in quantum transport of different topological phases in two-dimensional (graphene) and three-dimensional (Bi₂Se₃) TI nanoribbons under different configurations of the exchange field; (iv) non-magnetic- and magnetic-impurity-induced topological phase transitions in topological semimetals, with the prediction of a coexisting Dirac-Weyl mixed-phase under some given conditions. The results of this theoretical work in part elucidate some fundamental issues of magnetic topological materials, but also give indications for the experimental realization of quantum phenomena which have important technological applications.