Domain walls form naturally in the early Universe whenever a discrete symmetry is spontaneously broken at some phase transition. When each vacuum is populated equally, it is well known that the domain wall network comes to dominate the energy density of the Universe and causes excessive anisotropy in the cosmic microwave background. We present results for the initial conditions and dynamical evolution of domain wall networks in which one of the degenerate vacua has a population bias over the other. The initial distribution of domain walls is well described by percolation theory. We find that such networks, although they show evidence of a limited scaling regime for a range of biases, do not persist indefinitely. It follows that biased domain wall networks avoid the energy density and anisotropy problems. \textcopyright{} 1996 The American Physical Society.