Domain walls form naturally in the early Universe whenever a discrete
symmetry is spontaneously broken at some phase transition. When each v
acuum is populated equally, it is well known that the domain wall netw
ork comes to dominate the energy density of the Universe and causes ex
cessive anisotropy in the cosmic microwave background. We present resu
lts for the initial conditions and dynamical evolution of domain wall
networks in which one of the degenerate vacua has a population bias ov
er the other. The initial distribution of domain walls is well describ
ed by percolation theory. We find that such networks, although they sh
ow evidence of a limited scaling regime for a range of biases, do not
persist indefinitely. It follows that biased domain wall networks avoi
d the energy density and anisotropy problems.