This chapter’s objectives are to investigate mathematical programming problems with vanishing constraints employing quasidifferentiable functions, abbreviated QMPVC, and to establish appropriate optimality criteria. First, depending on the selection of quasidifferentials, we establish the Fritz-John (FJ) necessary conditions for optimality. We offer an appropriate form of the no nonzero abnormal multiplier constraint qualification (NNAMCQ-QMPVC) in order to obtain the Karush-Kuhn-Tucker (KKT) necessary criteria for optimality. We also suggest a few scenarios in which the selection of the quasidifferentials has no bearing on the Lagrange multipliers. With a suitable generalised convexity option, we also present sufficient optimality conditions that must be met for a weak stationary point of the QMPVC to be an optimal solution.