I review the physics of the Diffuse Supernova Neutrino flux (or Background, DSNB), in the context of future searches at the next generation of neutrino observatories. The theory of the DSNB is discussed in its fundamental elements, namely the cosmological rate of supernovae, neutrino production inside a core collapse supernova, redshift, and flavor oscillation effects. The current upper limits are also reviewed, and results are shown for the rates and energy distributions of the events expected at future liquid argon and liquid scintillator detectors of O(10) kt mass, and water Cherenkov detectors up to a 0.5 Mt mass. Perspectives are given on the significance of future observations of the DSNB, both at the discovery and precision phases, for the investigation of the physics of supernovae and of the properties of the neutrino.