Starting with general phenomenological considerations, a function describing the initiation as well as the stable and unstable growth of fatigue cracks is derived. In comparison with many other functions assigned to the description of fatigue crack growth curves, its merit lies in its compatibility with some currently used functions -in a stable growth region the general function turns into the Paris-Erdogan law and, if the initiation region is added, it turns into the modified Klesnil-Luka Β΄s Λrelation which enables a better fit of some experimental data than the original one. In the stable and subcritical regions the general function can turn into the Forman relation or into others, if special parameter values are chosen. Each parameters of the new function has a strictly defined geometrical meaning as far as the curve shape is concerned. A successful application of the function and its simplified forms for fitting other authors' experimental results for fatigue crack growth obtained from tests made with different metallic materials, is presented. Also the generalized form of the function for different cycle asymmetries is proposed and successfully verified.