First Hermann Weyl, then Julian Schwinger laid the foundations for a
systematic classification of quantum degrees of freedom and paved the way
toward a technical formulation of Bohr's Principle of Complementarity, on
which all of quantum kinematics is based. Central to the issue are
complementary observables and their mutually unbiased bases (MUB), which have attracted much interest in recent years. MUB are very useful for many quantum-information purposes. There are seemingly simple, but actually very
difficult, problems about MUB that have not been solved despite the efforts of
many. I will review the subject and report recent developments.
