  The paper aims to introduce the cyclotomic $q$-Schur superalgebra via the
permutation supermodules of the cyclotomic Hekce algebra and investigate its
structure. In particular, we show that the cyclotomic $q$-Schur superalgebra is
a cellular superalgebra and establish the Schur-Weyl duality between the
cyclotomic Hecke algebra and the cyclotomic $q$-Schur superalgebra.
