Progress in nuclear astrophysics by means of quantitative supernova spectroscopy is discussed with special concentration on type Ia, Ib and Ic and on SN 1987A. Spectral calculations continue to support an exploding C/O white dwarf as the best model of a SN Ia. Deflagration model W7 produces good maximum light spectra of SN Ia and seems to have a better composition distribution compared to delayed detonation models, but proper treatment of opacity remains a problem and the physical basis of SN Ia explosions is still not completely understood. All models for SN Ia predict large quantities of 56Co in the ejecta, but it is not clear that observations confirm this. Although the evolutionary origin of SN Ia remains uncertain, there is recent evidence that transfer of hydrogen in a binary system may be involved, as long suspected. There has been progress in comparing dynamical models with the optical/IR spectra of SN 1987A. The evolution of the [OI] λλ6300, 6364 feature and the presence of strong persistent HeI λ10 830 indicate that both the envelope and core material contribute substantially to the formation of emission lines in the nebular phase and that neither the core nor the envelope can be neglected. Blending with nearby hydrogen lines may affect both of these spectral features, thereby complicating the analysis of the lines. The effects of continuum transfer and photoionization have been included and are under study. The discrepancies between theoretical and observed spectra are due primarily to the one-dimensional hydrodynamic models. The spectral data are not consistent with the high density ``spike'' (in radial coordinate) of the core material that is predicted by all such models. Analysis of the light curves of SN Ib and SN Ic supernovae implies that there are significant differences in their physical properties. Some SN Ib have considerably more ejecta mass than SN Ic events. SN Ib require He-rich atmospheres to produce the observed strong optical lines of HeI somewhat after maximum. SN Ic events require a considerable depletion, if not absence, of helium. Calculations of the nebular phase after about 200 days show that the optical spectra of SN Ib/c will not reveal HeI even if helium is present. The spectra at that phase are rather insensitive to variations in the mass and composition. The similarity of the optical spectra of SN Ib and Ic events at late times thus does not mean that they are physically very similar. Observations of the HeI λ10 830 line could provide a good diagnostic of the atmospheric composition of Sn Ib and SN Ic.