This is partly a tutorial and partly a review paper (with a few original additions) on saturation curves (SC) which describe the dependence of fluorescence intensity on laser intensity in atomic spectroscopy. The interest in SCs stems from applications in analytical chemistry, plasma diagnostics, physical and chemical kinetics, etc., as well as from their fundamental implications. After a brief introduction, some general facts and basic assumptions regarding atom-laser interactions are critically examined (Section 2.1) and the concepts of the “ideal” SC and saturation parameter are defined (Section 2.2). In the following Sections 3–7 various effects are discussed that can distort the SC and shift the (apparent) saturation parameter. The effects of a spatially, a temporally and a spectrally inhomogeneous laser beam, of laser-enhanced chemical reactions and ionization processes, of an optically thick atomic vapour and of various non-steady-state processes are successively reviewed. Atom trapping and polarization effects on the SCs measured recently with an atomic Na beam in vacuo are reported and discussed in some detail. Also, some new observations at high resolution on the spectrum of pulsed and cw multimode dye lasers are reported. In Section 8 some general conclusions are drawn and warnings given, and the possible extension of the concept of SC to multiphoton and multistep excitation processes, as well as to optogalvanic spectroscopy, is suggested.