Understanding the elementary steps involved in a chemical reaction forms the cornerstone of physical chemistry research. One way to deepen this understanding is by studying chemical and physical processes using linear and nonlinear spectroscopic techniques. However, the outcomes of such experiments can be difficult to decipher due to the interweaving of several effects. Therefore, in order to help experimentalists to disentangle such spectra, the role of theorists is to develop efficient tools that are able to accurately describe molecular systems. The starting point of such tools is solving the time-dependent Schrödinger equation. In this thesis, after implementing geometric integrators, which are based on a combination of the split-operator algorithm and Magnus expansion, for the exact nonadiabatic quantum dynamics of a molecule interacting with a time-dependent electromagnetic field, we derive and implement these geometric integrators for the time-dependent perturbation theory, the Condon, rotating-wave, and ultrashort-pulse approximations, as well as every possible combination thereof. As verified in several model systems, these integrators exactly preserve the geometric invariants, and achieve an arbitrary prescribed order of accuracy in the time step and an exponential convergence in the grid spacing. We also explore in more detail the ultrashort-pulse approximation and derive an analytical expression for the combination with the time-dependent perturbation theory; this expression significantly accelerates numerical calculations. We show that in the limit of the zero pulse width, the d-pulse approximation is recovered. We illustrate the performance of the introduced approximations, using a three-dimensional model of pyrazine, in which it is essential to go beyond the d-pulse limit in order to describe the dynamics correctly. The high-order algorithms are also applied to the photodissociation dynamics of iodomethane (CH3I), following its excitation to the A band. We implement a general split-operator with both discrete-variable and finite-basis representations that can treat one non-Cartesian, such as angular coordinate. To test the effect of various degrees of freedom and of the nonadiabatic dynamics, we apply these algorithms to one-, two-, and three-dimensional models of iodomethane, both in the presence and in the absence of nonadiabatic couplings. A full quantum calculation is, however, limited to problems with low dimensionality (approximately ten degrees of freedom). Beyond this, one must seek an affordable balance between computational efficiency and physical accuracy and can employ, for example, semiclassical methods that are based on classical trajectories. A simple semiclassical approximation that can treat larger systems and requires only local knowledge of the potential is the on-the-fly ab initio thawed Gaussian approximation. We implement a generalization of the method that goes beyond the Franck-Condon approximation and treats Herzberg-Teller active molecules. Our method is used to compute absorption spectra of phenyl radical and of benzene, for which the Herzberg-Teller contribution is essential.