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Topological insulators and superconductors are gapped quantum phases characterized by robust boundary states rather than symmetry breaking. In this talk, I review recent progress in their classification, starting from free-fermion systems described by Dirac Hamiltonians. I then discuss how crystalline symmetries enrich the phase structure and lead to new topological phenomena. Finally, I present a unified and systematic framework based on the Atiyah–Hirzebruch spectral sequence that enables a practical classification of topological and geometrical invariants.