A new approach is used to examine the stability of feedback systems comprising a time-varying linear subsystem and a time-varying non-linearity. The paper examines three stability aspects of the systems dynamic response: boundedness, unboundedness, and asymptotic decay. In addition to qualitative criteria, the results provide explicit quantitative bounds on the system response. The distinct time-domain method of analysis presented stipulates rather weak a priori restrictions on the nature of the linear and non-linear parts of the system, thus admitting a relatively broad class of systems. In the general time-varying non-linear feedback system considered, the non-linearity may exhibit hysteresis, and the linear subsystem may be non-causal, and may comprise any number of feed-forward differentiators of any order. The criteria obtained are given a simple graphical interpretation. A transformation which trades time-varying gains between plant and non-linearity is introduced. A number of examples demonstrate that the present results can predict stability information not obtainable from other existing criteria. In one example of a system with stationary plant, the present criteria prove superior to the Circle Theorem.