Speaker: Rolf Walder Institut fur Astronomie, ETH-Zentrum (Swiss) "New Developments In Core-Collapse Supernovae Simulations" In a first part I give an overview of the state of the art supernova theory, which shall provide elementary insights to the non-specialists and recalls the basic questions. The overview provides the basic observational facts, theoretical ideas on what actually sets the explosion and lightcurve scale, discusses 3 ideas on the explosion mechanism in some detail, and reviews the significant computational difficulties. In the second part, I present new results from the first truly multi-dimensional neutrino-hydrodynamical simulations. We provide a new set of testable predictions, from neutron star masses, rotation rates and kick velocities, to asymmetric neutrino signals, and gravitational waveforms. I then discuss new ideas on the importance of neutrino-driven winds from the proto-neutron star. These winds accelerate up to 25 percent of the speed of light and carry in several seconds approximately 0.5 percent of a solar mass. They may play a significant role in setting the explosion energy scale and may provide conditions for r-process nucleosynthesis. Finally, I will outline future perspectives of the field and give a short review of a new generation of computational tools which will allow to study supernova theory in much more detail and will allow to make comparisons to observations on a quantitative level for the first time.