In this thesis I present three theoretical studies on novel phenomena arising in two-dimensional materials, which can be exploited to tailor their emergent topological properties. In graphene, I propose two schemes for engineering two independent topological phases of matter, both relying on its unconventional low-energy properties. On the one hand, I consider graphene deposited on a magnetic substrate as a possible detector for skyrmions, using as a working principle the anomalous Hall signal produced by these magnetic whirls. The linear semimetallic dispersion of graphene, together with its suitability for extrinsic engineering through proximity, render the detection especially effective. I then focus on a special quantum spin Hall phase with broken time-reversal symmetry that can be generated in twisted graphene bilayers in the quantum Hall state. In the regime of large twisting angles the interplay between electronic interactions and an applied electric field drives the system into the targeted phase, which can be used as a novel platform for topological superconductivity. Finally, I focus on a two-dimensional electron gas with strong spin-orbit coupling in the quantum Hall regime. The study of the topological properties of this system when it is proximitized with a superconducting strip and a Zeeman field is applied parallel to it reveals unconventional features that allow for a novel implementation of a topological π Josephson junction. This phase is relevant for implementing a universal set of protected single-qubit operations.
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