  The dynamical critical behavior of a single directed line driven in a random
medium near the depinning threshold is studied both analytically (by
renormalization group) and numerically, in the context of a Flux Line in a
Type-II superconductor with a bulk current $\vec J$. In the absence of
transverse fluctuations, the system reduces to recently studied models of
interface depinning. In most cases, the presence of transverse fluctuations are
found not to influence the critical exponents that describe longitudinal
correlations. For a manifold with $d=4-\epsilon$ internal dimensions,
longitudinal fluctuations in an isotropic medium are described by a roughness
exponent $\zeta_\parallel=\epsilon/3$ to all orders in $\epsilon$, and a
dynamical exponent $z_\parallel=2-2\epsilon/9+O(\epsilon^2)$. Transverse
fluctuations have a distinct and smaller roughness exponent
$\zeta_\perp=\zeta_\parallel-d/2$ for an isotropic medium. Furthermore, their
relaxation is much slower, characterized by a dynamical exponent
$z_\perp=z_\parallel+1/\nu$, where $\nu=1/(2-\zeta_\parallel)$ is the
correlation length exponent. The predicted exponents agree well with numerical
results for a flux line in three dimensions. As in the case of interface
depinning models, anisotropy leads to additional universality classes. A
nonzero Hall angle, which has no analogue in the interface models, also affects
the critical behavior.
