Core-collapse supernova explosions occur at the end of the life of massive stars. Nuclear fusion in the stellar core builds elements up to iron until the inner core becomes unstable to gravitational collapse. The collapse is halted when the atomic nuclei have merged to uniform nuclear matter. A short dynamical bounce leads to the formation of a standing shock wave that slowly expands its radius against the continued accretion of infalling outer stellar layers. The high-density matter at the center forms a new-born neutron star and may continue to collapse to a black hole if the accretion rate is too high. Otherwise, after a delay, an energetic explosion is thought to be launched above the surface of the neutron star producing ejecta with characteristic abundances of elements. Corresponding abundances of heavy elements are seen in the lightcurve of supernovae, but also in spectral lines of the next generation of stars that form out of the polluted interstellar gas. A confirmation of the collapse scenario has been obtained by the direct observation of neutrinos in the event of SN1987A. Neutrinos are copiously produced at high density and radiate away about 100 times the energy of the kinetic explosion during the collapse, accretion and neutron star cooling phases. The theoretical understanding of the supernova explosion mechanism is crucial for the understanding of the stellar life cycle, the feedback of internal energy to the interstellar gas in star-forming regions, and the enrichment of the Galaxy with heavy elements. Supernovae are active in all observational windows and emit a broad spectrum of electro-magnetic waves, neutrinos, cosmic rays and probably gravitational waves. A quantitative understanding of the supernova explosion mechanism may grant observational access to matter under extreme conditions not accessible in the laboratory where new physics could be discovered. In this project we build three-dimensional supernova models and aim to predict the emission of neutrinos and gravitational waves. In particular, we investigate the observable effects of an early QCD phase transition to quark matter in the neutron star. Moreover, we try to clarify the interaction of neutrinos, fluid instabilities and magnetic fields in the supernova explosion mechanism.