The primary objective of this article is twofold: to address the key concept of topology that impacts materials science in a major way and to convey the excitement to the materials community of recent significant advances in our understanding of the important topological notions in a wide class of materials with potential technological applications. A paradigm of topology/geometry → property → functionality is emerging that goes beyond the traditional microscopic structure → property → functionality relationship. The new approach delineates the active roles of topology and geometry in design, fabrication, characterization, and predictive modeling of novel materials properties and multifunctionalities. After introducing the essentials of topology and geometry, we elucidate these concepts through a gamut of nanocarbon allotropes of de novo carbons, hierarchical self-assembled soft- and biomaterials, supramolecular assemblies, and nanoporous materials. Applications of these topological materials range from sensing, energy storage/conversion, and catalysis to nanomedicine.
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