"This project intends to make groundbreaking progress towards the solution of one of the most pestering and long-standing riddles of stellar astrophysics, namely the question how massive stars explode as supernovae (SNe).State-of-the-art simulations in two dimensions (2D) now yield neutrino-powered (through underenergetic) explosions for a growing variety of progenitors and thus support the delayed neutrino-heating mechanism. However, sophisticated, fully self-consistent, 3D simulations are still lacking, the spherical symmetry of the progenitor star models is becoming a serious handicap, and better exploitation of observational constraints of the SN mechanism is urgently needed.For these reasons we plan a novel, comprehensive modeling approach, in which 3D hydrodynamics including all relevant microphysics will not only be employed for the launch phase of the SN blast wave by neutrino-energy deposition. Different from previous initiatives, 3D hydrodynamics will also be applied to the final stages of convective shell burning in the progenitor core before collapse in order to derive --for the first time-- self-consistent, multidimensional progenitor data for adopting them as initial conditions in the SN modeling. Moreover, the 3D explosion simulations will be continued consistently through the long-time evolution of the SN outburst into the gaseous remnant phase. This challenging approach promises fundamentally new insights into the processes that trigger and shape SN explosions and will revise our understanding of how SNe depend on the properties of their progenitor stars. Moreover, heading for a direct comparison of the derived theoretical models with nearby young SN remnants like Crab, Cassiopeia A, and SN 1987A, whose 3D morphology and composition are currently unfolded in stunning detail by multiwavelength observations, the project will lay the foundations of a powerful, innovative, and so far not exploited way of probing the physics deep inside the SN core."