This paper presents an all round review of recent deve-lopments in treating the singularities in boundary element methods bothfor numerical computing and for mathematical analysing.Approachesfor numerical treatment of singular and hyper-singular integrations arelisted.Singular behaviour of solution on non-smooth boundary are discus-sed and the mathematical tools for describing it

such as the Sobolevspaces defined on a part of boundary

the pseudo-differential operatorsare presented.In order to incorporate the singular behaviour into theboundary element approximation

the technique of introducing singularboundary element is suggested.