The ground and excited electronic states are responsible for several materials' properties. The modern capability of rapidly solving on a computer the fundamental equations of relativistic or semi-relativistic quantum mechanics allows to compute the electronic structure of realistic systems, and even to predict the behaviour of materials before they are actually made. An initial estimate on the atomistic structure is sufficient to start a first-principles simulation, where no empirical parameters are needed, and compute a variety of physical properties. Computational experiments can be performed to test or inspire ideas, validate new theories and interpret experimental data. In this context, geometrical and topological aspects have emerged as fundamental components of the electronic structure, leading to a deeper understanding and rationalisation of surprising phenomena. Nowadays, concepts like the quantum metric-curvature tensor and topological invariants complement traditional electronic-structure theory in describing the behaviour of electrons in the potential of the nuclei. In this thesis we study the electronic structure and its topology in the context of materials design and discovery. In particular, we begin by focusing on a specific class of materials, namely two-dimensional topological insulators. By coupling state-of-the-art first-principles simulations with materials' informatics we screen materials databases looking for novel two-dimensional topological insulators. In this search, we find novel candidates and provide a picture of the abundance of such materials in nature. In the process, we identify one outstanding candidate, jacutingaite, displaying fairly unique physical properties. We unveil the electronic structure of monolayer jacutingaite and show its strong links with that of graphene. Monolayer jacutingaite in fact realises the topological physics of graphene, but at a much higher and more relevant energy scale, while retaining a richer interplay between spin-orbit coupling, crystal-symmetry breaking and dielectric response, that is potentially relevant for applications. We also study jacutingaite in its layered three-dimensional bulk form. We show how bulk jacutingaite is a dual topological insulator, where a non-trivial coupling between graphene-like layers induces an additional topological crystalline order with protected (001)-surface states; this is confirmed by experiments. Finally, we also discuss how to distinguish metals from insulators locally in real space, by further developing a formalism based on the ground-state electronic distribution, in particular by allowing for its integration with first-principles simulations. In parallel, we also present some efforts for a more accurate and faster computational materials screening. We introduce a protocol to test precision and performance of pseudopotentials across the periodic table, leading to the development of the standard solid-state pseudopotential (SSSP) libraries. We also focus on automating the construction of Wannier functions, that are particular relevant in the study of electronic structure in general and topological properties in particular. Last, we briefly mention some work done in software development for community-driven scientific codes, that supports the research effort.