Ruben Cabezon - March 10, 2017 Core Collapse Supernovae in the hall of mirrors: the Smoothed Particle Hydrodynamics perspective Core-collapse Supernova (CCSN) pose a difficult problem in Computational Astrophysics due to their variety of dynamical ranges and scales, jointly with the entanglement of complex physics that have different relevance at different stages. In spite of this, the use of numerical simulations is nowadays our best tool to gain insight into the (still unknown) mechanism that triggers such energetic explosions and the formation of the most compact objects in the universe, namely, neutron stars and black holes. Although the realm of computational simulations of CCSN is vastly dominated by Eulerian codes, the use of a meshless Lagrangian approach might provide new tools to explore with high resolution the proto-neutron star and its properties. After a brief review on the CCSN scenario, I will show some of the latest improvements related to the Smoothed Particle Hydrodynamics (SPH) technique, that have helped to overcome various long-lasting problems of this numerical technique. I will also show the first results that we obtain when we use SPH jointly with a spectral neutrino treatment to simulate CCSN.