  We perform a systematic study of several models that have been proposed for
the purpose of understanding the motion of driven interfaces in disordered
media. We identify two distinct universality classes: (i) One of these,
referred to as directed percolation depinning (DPD), can be described by a
Langevin equation similar to the Kardar-Parisi-Zhang equation, but with
quenched disorder. (ii) The other, referred to as quenched Edwards-Wilkinson
(QEW), can be described by a Langevin equation similar to the Edwards-Wilkinson
equation but with quenched disorder. We find that for the DPD universality
class the coefficient $\lambda$ of the nonlinear term diverges at the depinning
transition, while for the QEW universality class either $\lambda = 0$ or
$\lambda \to 0$ as the depinning transition is approached. The identification
of the two universality classes allows us to better understand many of the
results previously obtained experimentally and numerically. However, we find
that some results cannot be understood in terms of the exponents obtained for
the two universality classes {\it at\/} the depinning transition. In order to
understand these remaining disagreements, we investigate the scaling properties
of models in each of the two universality classes {\it above\/} the depinning
transition. For the DPD universality class, we find for the roughness exponent
$\alpha_P = 0.63 \pm 0.03$ for the pinned phase, and $\alpha_M = 0.75 \pm 0.05$
for the moving phase. For the growth exponent, we find $\beta_P = 0.67 \pm
0.05$ for the pinned phase, and $\beta_M = 0.74 \pm 0.06$ for the moving phase.
Furthermore, we find an anomalous scaling of the prefactor of the width on the
driving force. A new exponent $\varphi_M = -0.12 \pm 0.06$, characterizing the
scaling of this prefactor, is shown to relate the values of the roughness
