Turbulent mixing induced by Rayleigh-Taylor (RT) instability occurs ubiquitously in many natural phenomena and engineering applications. As the simplest and primary descriptor of the mixing process, the evolution of mixing width of the mixing zone plays a notable role in the flows. The flows generally involve complex varying acceleration histories and widely varying density ratios, two dominant factors affecting the evolution of mixing width. However, no satisfactory theory for predicting the evolution has yet been established. Here a theory determining the evolution of mixing width in general RT flows is established to reproduce, first, all of the documented experiments conducted for diverse (i.e., constant, impulsive, oscillating, decreasing, increasing, and complex) acceleration histories and all density ratios. The theory is established in terms of the conservation principle, with special consideration given to the asymmetry of the volume-averaged density fields occurring in actual flows. The results reveal the sensitivity or insensitivity of the evolution of a mixing front of a neighboring light or heavy fluid to the degree of asymmetry and thus explain the distinct evolutions in two experiments with the same configurations.