Absorption spectroscopy (AS) represents a reliable method for the characterization of cold atmospheric pressure plasma jets. The method's simplicity stands out in comparison to competing diagnostic techniques. AS is an in situ, non-invasive technique giving absolute densities, free of calibration procedures, which other diagnostics, such as laser-induced fluorescence or optical emission spectroscopy, have to rely on. Ground state densities can be determined without the knowledge of the influence of collisional quenching. Therefore, absolute densities determined by absorption spectroscopy can be taken as calibration for other methods. In this paper, fundamentals of absorption spectroscopy are presented as an entrance to the topic. In the second part of the manuscript, a review of AS performed on cold atmospheric pressure plasma jets, as they are used e.g. in the field of plasma medicine, is presented. The focus is set on special techniques overcoming not only the drawback of spectrally overlapping absorbing species, but also the line-of-sight densities that AS usually provides or the necessity of sufficiently long absorption lengths. Where references are not available for measurements on cold atmospheric pressure plasma jets, other plasma sources including low-pressure plasmas are taken as an example to give suggestions for possible approaches. The final part is a table summarizing examples of absorption spectroscopic measurements on cold atmospheric pressure plasma jets. With this, the paper provides a 'best practice' guideline and gives a compendium of works by groups performing absorption spectroscopy on cold atmospheric pressure plasma jets.