Topological principles constitute at present an integral component of condensed matter physics, permeating the modern characterization of electronic states while also guiding materials design. In this brief Perspective, I highlight three research threads in single-particle topological band theory that have recently gained momentum: ($i$) the rise of the quantum geometric tensor, whose components can at present be directly accessed with optical probes; ($\mathit{ii}$) the notions of delicate and multigap topology, which fall outside the scope of tenfold way and symmetry-based indicators yet leave robust physical fingerprints; and (iii) the consideration of bundle gerbes, which capture formerly overlooked higher-form topological aspects of energy bands. These distinct directions have been elegantly woven together: delicate and multigap topological insulators have peculiar features in quantum geometry that can be conveniently captured by bundle gerbes. This viewpoint exposes the recently identified quantization of a nonlinear optical response, and it invites deeper and systematic investigations into geometric and topological aspects of band structures beyond conventional Berryology.

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• Electronic structure
• Geometric & topological phases
• Photoconductivity
• Shift current
• Topological insulators
• Topological materials
• Gauge symmetries
• Geometry
• Optical techniques
• Photoemission spectroscopy
• Topology