first extensive search for Pop. III stars has been performed early by @Bond, who discovered only a few stars with metallicity ${\rm [Fe/H]} =-3$ (i.e. a iron content of only 1/1000 of the solar value); more recently the sample of extremely metal-poor stars has significantly been increased, mainly due to extensive Wide-Field Objective-Prism Surveys, and especially due to the Preston-Shectman survey [@BPS92 usually also called HK survey]. Although none of these stars shows a  Pop. III chemical composition, there is an increasing number of stars with metallicity ${\rm [Fe/H]} <-3$, for which accurate post-detection analysis can be performed. In addition, one can expect that forthcoming high resolution spectroscopy from the new generation of $10\,$m-class telescopes will lead to a big improvement in the search for this class of peculiar objects. Undoubtedly, unprecedented results in this field will be achieved with planned space-based survey instruments such as GAIA.

In order to fully understand the results of these surveys, it is necessary to investigate theoretically not only the main evolutionary properties of low-mass metal-free stars, but also the changes of the surface chemical composition appearing during that evolution. These may be caused by accretion of metal-rich matter from the interstellar medium and/or 