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In the mid-2000s, a new quantum state of topological insulators was proposed. It deeply refreshed the traditional understanding of electronic band structure, which has been the most fundamental tool to classify metals and insulators. Topological insulators with non-trivial topological charges can host robust surface states or edge states located in the bulk bandgap. To understand this new state, an understanding of the bandgap is not sufficient, and it led to the new field of topological band theory in condensed matter physics. The development of electronic band structure theory also inspired the understanding of topological band theory from the chemical point of view and results in the new topic of topological chemistry. The discovery of topological insulators motivated extensive studies of solid-state materials from topological theory, leading to many topological materials in both insulators and metals. In the last 15 years, various topological materials characterized by different topological electronic structures have been discovered. One of the most important features shared by all different topological materials is the topologically protected non-trivial surface states (TSSs). Such TSSs are essentially different from the dangling bonds because they connect to conduction bands and valence bands in insulators or bulk band crossings in metals. The extra perturbation can only change their detailed shape but not remove them. This characteristic makes TSSs attractive for practical applications in the quantum information process, data storage, and energy conversion. In particular, the robust surface state is an attractive property that benefits energy-related catalysis. The last few years have seen research in this field with a focus on developing efficient topological material catalysts for hydrogen evolution reaction (HER), oxygen evolution reaction (OER), and reduction. To date, the topological catalyst has become a new frontier in both chemistry and materials science. Within the scope of this Ph.D. thesis, several topological semimetals and their HER activity are studied with the help of density functional theory, electrochemical theory, and topological band theory, combined with experimental measurements performed within the workgroup. The spectrum of performed projects ranges from the theoretical design of the high-efficiency hydrogen evolution catalyst with the guidance of topology in close collaboration with experiments and in-depth understanding of the relationship between topological properties and catalysis.

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