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Quantum mechanical systems with some degree of complexity due to multiplescattering behave as if their Hamiltonians were random matrices. Such behavior,while originally surmised for the interacting many-body system of highlyexcited atomic nuclei, was later discovered in a variety of situationsincluding single-particle systems with disorder or chaos. A fascinating themein this context is the emergence of universal laws for the fluctuations ofenergy spectra and transport observables. After an introduction to the basicphenomenology, the talk highlights the role of symmetries for universality, inparticular the correspondence between symmetry classes and symmetric spacesthat led to a classification scheme dubbed the 'Tenfold Way'. Perhapssurprisingly, the same scheme has turned out to organize also the world oftopological insulators.

Of symmetries, symmetry classes, and symmetric spaces: from disorder and quantum chaos to topological insulators