Cutting-edge issues in core-collapse supernova theory

Based on our multi-dimensional neutrino-radiation hydrodynamic simulations, we report several cutting-edge issues about the long-veiled explosion mechanism of core-collapse supernovae (CCSNe). In this contribution, we pay particular attention to whether three-dimensional (3D) hydrodynamics and/or general relativity (GR) would or would not help the onset of explosions. Our results from the first generation of full GR 3D simulations including approximate neutrino transport are quite optimistic, indicating that both of the two ingredients can foster neutrino-driven explosions. We give an outlook with a summary of the most urgent tasks to draw a robust conclusion to our findings.