We begin with a brief critical discussion of the existing definitions of melting and damage thresholds and the different kinds of experimental determinations of the thresholds. Then we investigate the thermal and athermal melting of a wide-band gap semiconductor (SiO2) and of silicon by two different methods consisting of a rate equation for the excited electrons and of a complete self-consistent solution of a coupled system of differential equations for the electron density and for the electron and phonon temperatures. In particular, we focus on the role of the critical electron density in the case of athermal melting. Our calculations suggest that this is determined by the critical plasma density. Finally, we conclude with a discussion of the benefits and drawbacks of the two approaches.