Exceptional points are singularities that occur generically in the spectrum and eigenfunctions of operators (usually matrices) that depend on a parameter. For selfadjoint operators they always lie in the complex plane of the parameter. Owing to their association with level repulsion they feature prominently in quantum chaos. Being singularities they play an essential role in a great variety of approximation schemes. Recent experiments have nicely illustrated the Riemann structure of the singularities (square root type for the energies and fourth root type for the state vectors) as well as the chiral character of the state vectors at the singularities. The latter property depends crucially on the particular type of the operator considered.