In this paper the fatigue strength of U-shaped bellows under axial loading is studied. Fatigue tests of bellows in the elastic and plastic ranges are carried out by using the fatigue testing machine designed for this research, and the results are compared with the fatigue strength of test pieces made of the same material as bellows. From experimental data a formula for estimation of the life of U-shaped bellows made of stainless steel is proposed, i.e. [numerical formula] where N is cycles to failure and σ is the maximum stress value calculated from the design formula or charts in the 1st or 2nd report. This formula is compared with M.W.Kellogg's formula which is nowadays usually employed.