ESDU 84009 introduces the use of Describing Functions (DF) to estimate the stability and the steady-state response of autonomous closed-loop non-linear systems. The DF uses a single sinusoidal input to the non-linear element and is therefore classified as a frequency response method. Five DF schemes are given and the characteristics of those DF's when used with eight common non-linear elements (combinations of deadzone, saturation, ideal relay, hysteresis and backlash) defined graphically and by their algebraic expressions. Two practical worked examples of third-order non-linear systems illustrate the use of those DF's in stability investigations and show the accuracy with which they estimate the magnitude of the output. Those DF's are also used to estimate the frequency and magnitude of a self-sustaining, or limit-cycle, oscillation.