The dynamics of driven interfaces in the random-field Ising model (RFIM) is investigated by the use of Monte Carlo simulations. Interfaces in the RFIM separate regions of opposite spin orientation. By applying an external field one orientation is energetically favored. This may yield an interface motion which is hindered by the random-field. Without thermal fluctuations the competition between the driving field and the random-field leads to a so-called depinning transition. A permanent interface motion is found only if the driving field exceeds a threshold field H_c. At the transition point the interface velocity vanishes continuously, characterized by a critical exponent beta. The values of beta found in the RFIM for the dimensions d=3,4,5,6 support the assumption that the depinning transition in the RFIM belongs to the universality class of the Edwards-Wilkinson equation with quenched disorder. The energy barriers which cause a pinning of the interface at temperature T=0 can be overcome due to the energy provided by thermal fluctuations. This yields a permanent interface motion. For sufficient small driving fields a so-called creep regime is found in the random-field Ising model. This creep regime is predicted by phenomenological theories, functional renormalization group calculations, and has been observed in experiments. The field dependence of the energy barrier in the RFIM is investigated and the results are compared with those known in the literature. Furthermore, it is investigated whether the influence of temperature on the depinning transition can be understood within the theory of critical phenomena. It is assumed that the interface velocity can be expressed as a generalized homogenous function in the vicinity of the transition point (H=H_c|T=0). This assumption is supported by the results of simulations in the dimensions d=3,4,6, yielding an algebraic decay v(H=H_c) proportional T^(1/psi) with an exponent psi>0. The assumption of the interface velocity being a generalized homogenous function is also validated by simulations of magnetic films. From these simulations it can additionally be concluded that the depinning transition in magnetic films is characterized by the two dimensional exponents. The investigations of the five dimensional model show the occurrence of logarithmic correction revealing that d_c=5 is the upper critical dimension of the depinning transition in the RFIM.