A unified treatment of both superconformal and quasisuperconformal algebras with quadratic non-linearity is given. General formulas describing their structure are found by solving the Jacobi identities. A complete classification of quasisuperconformal and ${ \bf Z}_2\times{ \bf Z}_2$-graded algebras are obtained and in addition to the previously known cases five exceotional quasisuperconformal algebras and a series of ${ \bf Z}_2\times{ \bf Z}_2$-superconformal algebras containing affine $\widehat{ sp}_2\otimes\widehat{ osp}(N|2M)$ are constructed.
Categories: hep-th