The authors analyze the optimal replacement of a single asset under continuous and discontinuous technological change. They consider an infinite-horizon replacement problem with a variable asset lifetime. Due to deterioration, maintenance costs increase when the asset becomes older. Because of technological change, both maintenance and new capital costs decrease in time for a fixed asset age. The dynamics of the optimal asset lifetime are studied analytically and numerically in the cases of one and several technological breakthroughs. The breakthroughs cause irregularities (anticipation echoes) in the asset lifetime before the breakthrough time. It is shown that the optimal lifetime is always smaller when new capital costs decrease faster, when the maintenance costs decrease slower, or when both maintenance and new capital costs decrease faster with the same rate.