In this paper, we present detailed quantum treatment of the semirigid vibrating rotor target (SVRT) model for reaction dynamics involving polyatomic molecules. In the SVRT model, the reacting (target) molecule is treated as a semirigid vibrating rotor which can be considered as a three-dimensional generalization of the diatomic molecule. This model provides a realistic framework to treat reaction dynamics of polyatomic systems. Using the SVRT model, it becomes computationally practical to carry out quantitatively accurate quantum dynamics calculation for a variety of dynamics problems in which the reacting molecule is a polyatomic or complex molecule. In this work, specific theoretical treatment and mathematical formulation of the SVRT model are presented for three general classes of reaction systems: (1) reaction of an atom with a polyatomic molecule (atom-polyatom reaction), (2) reaction between two polyatomic molecules (polyatom-polyatom reaction), and (3) polyatomic reaction with a rigid surface (polyatom-surface reaction). Since the number of dynamical degrees of freedom in the SVRT model for the above three classes of dynamical problems is limited, accurate quantum (both ab initio and dynamical) calculations are possible for many reactions of practical chemical interest. In this paper, a time-dependent wave packet approach is employed to implement the SVRT model for dynamics calculation of polyatomic reactions.