[en] Symmetry-breaking phase transitions may leave behind topological defects with a density dependence on the quench rate. We investigate the dynamics of such quenches for the one-dimensional, Landau-Ginzburg case and show that the density of kinks, n, scales differently with the quench time scale, τ_Q, depending on whether the dynamics in the vicinity of the critical point is overdamped (n∝τ_Q^-1/4) or underdamped (n∝τ_Q^-1/3). Either of these cases may be relevant to the early Universe, and we derive bounds on the initial density of topological defects in cosmological phase transitions. copyright 1998 The American Physical Society