A singularity is said to be weakly-exceptional if it has a unique purely log terminal blow up. In dimension 2, V. Shokurov proved that weakly exceptional quotient singularities are exactly those of types Dn, E6, E7, E8. This thesis classifies the weakly exceptional quotient singularities in dimensions 3, 4 and 5, and proves that in any prime dimension, all but finitely many irreducible groups give rise to weakly exceptional singularities. It goes on to provide an algorithm that produces such a classification in any given prime dimension.