  Quantum mechanical systems with some degree of complexity due to multiple
scattering behave as if their Hamiltonians were random matrices. Such behavior,
while originally surmised for the interacting many-body system of highly
excited atomic nuclei, was later discovered in a variety of situations
including single-particle systems with disorder or chaos. A fascinating theme
in this context is the emergence of universal laws for the fluctuations of
energy spectra and transport observables. After an introduction to the basic
phenomenology, the talk highlights the role of symmetries for universality, in
particular the correspondence between symmetry classes and symmetric spaces
that led to a classification scheme dubbed the 'Tenfold Way'. Perhaps
surprisingly, the same scheme has turned out to organize also the world of
topological insulators.
