The axion arises as a consequence of the spontaneous breaking of Peccei-Quinn (PQ) symmetry, which is a viable solution to the strong CP problem of quantum chromodynamics and may explain the dark matter of the universe. However, this PQ mechanism predicts the formation of topological defects such as strings and domain walls in the early universe, and the evolution and decay of these defects give some implications for the early universe cosmology. In particular, the analysis of the spectrum of axions radiated from networks of topological defects revealed that axions produced by the defects give significant contributions to the relic cold dark matter abundance. In this talk, I will report recent results of field-theoretic lattice simulations of topological defects and discuss cosmological constraints on axion models.