Electronic spin is one of the most fundamental observables in quantum mechanics, and in recent years, spin polarization in organic systems has received a great deal of interest. In this thesis, we investigate a novel mechanism for coupling together spin and nuclear dynamics, namely the molecular Berry curvature effects. In the first part of the thesis, we demonstrate that the molecular Berry curvature effects can lead to spin-specific reaction rates and spin-dependent reaction pathway selection. In systems with odd number of electrons and spin-orbit coupling, the nuclei will experience a Lorentz-like force from the Berry curvature on the adiabatic surfaces. Such a ``Berry force'' depends on the electronic spin direction and therefore can lead to spin-dependent nuclear motion. By using scattering calculations in two-dimensional model systems, we find that the reaction channel selectivity can reach unity even in the presence of a moderate spin-orbit coupling. In the second part of the thesis, we propose a few extensions to Tully's fewest switches surface hopping in complex-valued Hamiltonians. The most significant and successful extension is the pseudo-diabatic phase-space surface hopping (PD-PSSH), which we adapt from Shenvi's PSSH algorithm. We find that our PD-PSSH algorithm can correctly capture molecular Berry curvature effects in complex-valued Hamiltonians and achieve a well-defined rescaling direction, while retaining a simple and intuitive algorithmic form. Moreover, we derive and analyze a preconditioned quantum-classical Liouville equation (QCLE) theory that directly connects to the PD-PSSH algorithm and explains the latter's success in simulating complex-valued Hamiltonians. We expect that with the PD-PSSH algorithm, we can explore the coupled spin-nuclear nonadiabatic dynamics in real systems, which may be abundant in systems with spin-orbit coupling or external magnetic fields.