Perturbation theory in curved spacetimes is well studied for bosonic fields and spin-1/2 fields in various 4-dimensional and higher dimensional maximally symmetric spacetimes, but not for spin-3/2 fields. The main motivation of our work is filling this gap. We first derive a complete set of spinor-vector eigenmodes on the N-sphere. Next we consider the gauge symmetry and define the supercovariant derivative for the spin-3/2 fields in D=N+2 dimesional maximally symmetric spacetimes. Using the supercovariant derivative to obtain the equations of motion, separating the angular and radial-time components by the spinor-vector eigenmodes, and finally obtain the master radial equation for this study. With the radial equation, a study of the quasi-normal modes by using the semi-analytic methods will be presented as the last part of this presentation.