The question why and how core-collapse supernovae (SNe) explode is one of the central and most long-standing riddles of stellar astrophysics. Solving this problem is crucial for deciphering the SN phenomenon, for predicting its observable signals such as light curves and spectra, nucleosynthesis yields, neutrinos, and gravitational waves, for defining the role of SNe in the dynamical and chemo-dynamical evolution of galaxies, and for explaining the birth conditions and properties of neutron stars (NSs) and stellar-mass black holes. Since the formation of such compact remnants releases over hundred times more energy in neutrinos than the kinetic energy of the SN explosion, neutrinos can be the decisive agents for powering the SN outburst. According to the standard paradigm of the neutrino-driven mechanism, the energy transfer by the intense neutrino flux to the medium behind the stagnating core-bounce shock, assisted by violent hydrodynamic mass motions (sometimes subsumed by the term "turbulence"), revives the outward shock motion and thus initiates the SN explosion. Because of the weak coupling of neutrinos in the region of this energy deposition, detailed, multi-dimensional hydrodynamic models including neutrino transport and a wide variety of physics are needed to assess the viability of the mechanism. Owing to advanced numerical codes and increasing supercomputer power, considerable progress has been achieved in our understanding of the physical processes that have to act in concert for the success of neutrino-driven explosions. First studies begin to reveal observational implications and avenues to test the theoretical picture by data from individual SNe and SN remnants but also from population-integrated observables. While models will be further refined, a real breakthrough is expected through the next Galactic core-collapse SN, when neutrinos and gravitational waves can be used to probe the conditions deep inside the dying star.