BRST Operator for Superconformal Algebras with Quadratic Nonlinearity

We construct the quantum BRST operators for a large class of superconformal and quasi--superconformal algebras with quadratic nonlinearity. The only free parameter in these algebras is the level of the (super) Kac-Moody sector. The nilpotency of the quantum BRST operator imposes a condition on the level. We find this condition for (quasi) superconformal algebras with a Kac-Moody sector based on a simple Lie algebra and for the $Z_2\times Z_2$--graded superconformal algebras with a Kac-Mody sector based on the superalgebra $osp(N\vert 2M)$ or $s\ell(N+2\vert N)$.