Massive stars live fast and die young. They shine furiously for a few million years, during which time they synthesize most of the heavy elements in the universe in their cores. They end by blowing themselves up in a powerful explosion known as a supernova (SN). During this process, the core collapses to a neutron star or a black hole, while the outer layers are expelled with velocities of thousands of kilometers per second. The resulting fireworks often outshine the entire host galaxy for many weeks. The explosion energy is eventually radiated away, but powering of the newborn nebula continues by radioactive isotopes synthesized in the explosion. The ejecta are now quite transparent, and we can see the material produced in the deep interiors of the star. To interpret the observations, detailed spectral modeling is needed. This thesis aims to develop and apply state-of-the-art computational tools for interpreting and modeling SN observations in the nebular phase. This requires calculation of the physical conditions throughout the nebula, including non-thermal processes from the radioactivity, thermal and statistical equilibrium, as well as radiative transport. The inclusion of multiline radiative transfer, which we compute with a Monte Carlo technique, represents one of the major advancements presented in this thesis. On February 23 1987, the first SN observable by the naked eye since 1604 exploded, SN 1987A. Its proximity has allowed unprecedented observations, which in turn have lead to significant advancements in our understanding of SN explosions. As a first application of our model, we analyze the 44Tipowered phase (t & 5 years) of SN 1987A. We find that a magnetic field is present in the nebula, trapping the positrons that provide the energy input, and resulting in strong iron lines in the spectrum. We determine the 44Ti mass to 1.5(+0.5−0.5)*10−4 M⊙. From the near-infrared spectrum at an age of 19 years, we identify strong emission lines from explosively synthesized metals such as silicon, calcium, and iron. We use integral-field spectroscopy to construct three-dimensional maps of the ejecta, showing a morphology suggesting an asymmetric explosion. The model is then applied to the close-by and well-observed Type IIP SN 2004et, analyzing its ultraviolet to mid-infrared evolution. Based on its Mg I] 4571 Å, Na I 5890, 5896 Å, [O I] 6300, 6364 Å, and [Ne II] 12.81 mm nebular emission lines, we determine its progenitor mass to be around 15 M⊙. We confirm that silicate dust, SiO, and CO have formed in the ejecta. Finally, the major optical emission lines in a sample of Type IIP SNe areanalyzed.We find that most spectral regions in Type IIP SNe are dominated by emission from the massive hydrogen envelope, which explains the relatively small variation seen in the sample. We also show that the similar line profiles seen from all elements suggest extensive mixing occurring in most hydrogenrich SNe. / At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 3: Manuscript. Paper 4: Accepted.