Exceptional two-dimensional singularities
are singularities of type E_6, E_7 and E_8.
Exceptional three-dimensional quotient singularities
have been classified by Yura Prokhorov and Dima Markushevich.
We show how to use the idea of Koll\'ar to prove that
exceptional singularities exist in all dimensions.
We show how to use the Nadel-Shokurov vanishing
with the Kawamata subadjunction to classify exceptional
quotient singularities in dimensions 4,5,6 and 7.
This is joint work with Kostya Shramov.