  We construct the covariant $\kappa$-symmetric superstring action for type
$IIB$ superstring on plane wave space supported by Ramond-Ramond background.
The action is defined as a 2d sigma-model on the coset superspace. We fix the
fermionic and bosonic light-cone gauges in the covariant Green-Schwarz
superstring action and find the light-cone string Lagrangian and the
Hamiltonian. The resulting light-cone gauge action is quadratic in both the
bosonic and fermionic superstring 2d fields, and therefore, this model can be
explicitly quantized. We also obtain a realization of the generators of the
basic superalgebra in terms of the superstring 2d fields in the light-cone
gauge.
