Spin dynamics in pulsed Nuclear Magnetic Resonance (NMR) spectroscopy are studied theoretically using spin density operator formulation. The theory exploits the use of irreducible spherical tensor (multipole) operators of spin angular momentum in developing alternative treatments to spin dynamical problems. Applications in spin decoupling and soft pulse dynamics are considered, giving new analytical solutions for the complete density matrix. The focus of the thesis, however, is on the development of two distinctly new methods of solution to the quantum Liouville equation for NMR field modulation. The first method derives a linear integral equation for the magnetization and provides a rapidly converging integral series solution which is useful in explaining many pulse shapes. The second method uses the Wei-Norman Lie algebra approach$ sp{ rm 96}$ which reduces the complete arbitrary spin I problem to a Riccati equation. Novel exact analytical solutions for a class of temporal field modulations are obtained. In contrast to all other existing formulations, the theory provides the first known examples of exactly soluble nonrectangular pulse shapes for all resonance off-set, all field amplitudes, all time and valid for all spin I $ geq$ 1/2. In addition, a new series expansion for the complete spin density matrix is obtained. The theory is applied to producing spin inversion profiles which show applicability to selective excitation.