The dynamics of protons, the nuclei of hydrogen atoms, are of focal interest in chemistry and biophysics. This research encompasses fundamental theoretical aspects of the proton-transport dynamics: the spectral signatures, the reaction kinetics and the non-Markovian nature. The transfer motion of the excess proton between two acceptors, such as water itself, is not a normal mode. It is rather described, according to reaction-rate theory as stochastic motion over an energy barrier. The corresponding time scales are extracted from ab initio simulations and related to the spectra via an analytical model. Secondly, another approach for modeling spectroscopic signatures beyond the harmonic normal-mode approximation is introduced: the inclusion of anharmonic potential and frequency-dependent friction effects. The generalized Langevin equation presents an effective dynamical model, which handles arbitrary nonharmonic potential shapes in conjunction with non-Markovian time-dependent friction, and is here employed to extend the normal-mode-based description of vibrational spectra and applied to ab initio simulation data of water. Finally, some aspects of reaction-rate theory are discussed, that follow from such non-Markovian memory friction models, and presumably are relevant for proton-transfer reactions. Besides the memory friction, the effective mass and the potential shape are relevant variables of theoretical reaction-rate models. A perspective on the competition of these three effects on reaction times is demonstrated for pair-dissociation dynamics in water from classical molecular-dynamics simulations.