There exist very few results on mixing for non-stationary processes. However, mixing is often required in statistical inference for non-stationary processes such as time-varying ARCH (tvARCH) models. In this paper, bounds for the mixing rates of a stochastic process are derived in terms of the conditional densities of the process. These bounds are used to obtain the α, 2-mixing and β-mixing rates of the non-stationary time-varying ARCH(p) process and ARCH(∞) process. It is shown that the mixing rate of the time-varying ARCH(p) process is geometric, whereas the bound on the mixing rate of the ARCH(∞) process depends on the rate of decay of the ARCH(∞) parameters. We note that the methodology given in this paper is applicable to other processes.