We review the recent progress in understanding the properties of spin-split superconductors under non-equilibrium conditions. Recent experiments and theories demonstrate a rich variety of transport phenomena occurring in devices based on such materials that suggest direct applications in thermoelectricity, low-dissipative spintronics, radiation detection and sensing. We discuss different experimental situations and present a theoretical framework based on quantum kinetic equations. Within this framework we provide an accurate description of the non-equilibrium distribution of charge, spin and energy, which are the relevant non-equilibrium modes, in different hybrid structures. We also review experiments on spin-split superconductors and show how transport measurements reveal the properties of the non-equilibrium modes and their mutual coupling. We discuss in detail spin injection and diffusion and very large thermoelectric effects in spin-split superconductors.

Version published in Rev. Mod. Phys. Long version to be published separately elsewhere