Fatigue damage in a system with one degree of freedom is one of the two criteria applied for the comparison of severity of many vibratory environments. This criterion is also employed for a specification representing the effects produced by the set of vibrations imposed in useful life. In this volume, which is devoted to the calculation of fatigue damage, the hypotheses adopted to describe the behaviour of material suffering fatigue and the laws of fatigue accumulation are explored. Also considered are the methods of counting the response peaks, which are used to establish the histogram when it is impossible to use the probability density of the peaks obtained with a gaussian signal. The expressions for means damage and its standard deviation are established, and other hypotheses tested.