Abstract The concepts involved in the construction and interpretation of inverse isochron diagrams used in 40Ar/ 39Ar geochronology are reviewed. The diagrams can be useful as a means of recognising atmospheric argon and excess 40Ar, incorporated in the mineral lattice, which cannot be recognised from 40Ar/ 39Ar spectra. The age established using an inverse isochron plot, unlike that yielded by a spectrum, is not affected by trapped argon 40Ar/ 36Ar ratios that are different from the atmospheric argon ratio (e.g. due to excess 40Ar), and may contribute to a better age interpretation. However, a heterogeneous distribution of excess 40Ar or heterogeneous argon loss can cause ‘false’ isochrons, with axial intercepts indicating an incorrect age and an incorrect trapped argon composition. Inconsistency between the ages from a spectrum and from the associated inverse isochron plot may indicate the degree of inaccuracy of isochrons. However, it is possible that both the spectrum and inverse isochron yield the same incorrect age. The importance of considering all possible interpretations before assigning an age to a specimen is stressed.