We consider the morphology and dynamics of an interface driven through
a random two-dimensional medium by an applied force f. The onset of m
otion is a critical phenomenon, with mean velocity upsilon approximate
ly (f - f(c))zeta above the threshold force f(c). Fluctuations in the
velocity exhibit a power law noise spectrum. At large length scales th
e moving interfaces are self-affine with roughness exponent alpha = 0.
5. There is a crossover to different scaling behavior below the correl
ation length, xi approximately (f - f(c))-nu. The type of scaling at s
mall lengths depends upon the nature and strength of the disorder. Two
examples are considered - a magnetic domain wall model exhibiting sel
f-similar structure characteristic of percolation, and a fluid invasio
n model which produces self-affine scaling.