This dissertation deals with the dynamics of interacting quantum and classical spin models and the question of whether and to which degree the dynamics of these models agree with each other. For this purpose, XXZ models are studied on different lattice geometries of finite size, ranging from one-dimensional chains and quasi-one-dimensional ladders to two-dimensional square lattices. Particular attention is paid to the high-temperature analysis of the temporal behavior of autocorrelation functions for both the local density of magnetization (spin) and energy, which are closely related to transport properties of the considered models. Due to the conservation of total energy and total magnetization, the dynamics of such densities are expected to exhibit hydrodynamic behavior for long times, which manifests itself in a power-law tail of the autocorrelation function in time. From a quantum mechanical point of view, the calculation of these autocorrelation functions requires solving the linear Schrödinger equation, while classically Hamilton’s equations of motion need to be solved. An efficient numerical pure-state approach based on the concept of typicality enables circumventing the costly numerical method of exact diagonalization and to treat quantum autocorrelation functions with up to N = 36 lattice sites in total. While, in full generality, a quantitative agreement between quantum and classical dy- namics can not be expected, contrarily, based on large-scale numerical results, it is demonstrated that the dynamics of the quantum S = 1/2 and classical spins coincide, not only qualitatively, but even quantitatively, to a remarkably high level of accuracy for all considered lattice geometries. The agreement particularly is found to be best in the case of nonintegrable quantum models (quasi-one-dimensional and two-dimensional lattice), but still satisfactory in the case of integrable chains, at least if transport properties are not dominated by the extensive number of conservation laws. Additionally, in the context of disordered spin chains, such an agreement of the dynamics is found to hold even in the presence of small values of disorder, while at strong disorder the agreement is pronounced most for larger spin quantum numbers. Finally, it is shown that a putative many-body localization transition within the one- dimensional spin chain is shifted to stronger values of disorder with increasing spin quantum number. It is concluded that classical or semiclassical simulations might provide a meaningful strategy to investigate the quantum dynamics of strongly interacting quantum spin models, even if the spin quantum number is small and far from the classical limit.