The aim of this paper is to present the case that stellar mass primordial black holes make up the dark matter component of the Universe. A near critical density of compact bodies implies that most lines of sight will be gravitationally microlensed, and the paper focuses on looking for the predicted effects on quasar brightness and spectral variations. These signatures of microlensing include the shape of the Fourier power spectrum of the light curves, near achromatic and statistically symmetric variations, and the absence of time dilation in the time-scale of variability. For spectral changes it is predicted that as the continuum varies there is little corresponding change in the strength of the broad lines. In all these cases, the observations are found to be consistent with the predictions for microlensing by a population of stellar mass compact bodies. For multiply lensed quasar systems where the images vary independently and microlensing is the generally accepted explanation, the case is made that stellar populations are too small to produce the observed effects, and that the only plausible alternative is a population of compact dark matter bodies of around a stellar mass along the line of sight. The most serious objection to dark matter in the form of compact bodies has come from observations of microlensing of stars in the Magellanic clouds. In this paper, the expected event rate is re-analysed using more recent values for the structure and dynamics of the Galactic halo, and it is shown that there is then no conflict with the observations. Finally, the possible identity of a near critical density of dark matter in the form of stellar mass compact bodies is reviewed, with the conclusion that by far the most plausible candidates are primordial black holes formed during the QCD epoch. The overall conclusion of the paper is that primordial black holes should be seen alongside elementary particles as viable dark matter candidates.