The long-range Fröhlich electron–phonon interaction has been identified as the most essential for pairing in high-temperature superconductors owing to poor screening, as is now confirmed by optical, isotope substitution, recent photoemission and some other measurements. I argue that low-energy physics in cuprate superconductors is that of superlight small bipolarons, which are real-space hole pairs dressed by phonons in doped charge-transfer Mott insulators. They are itinerant quasiparticles existing in the Bloch states at low temperatures as also confirmed by the continuous-time quantum Monte–Carlo algorithm (CTQMC) fully taking into account realistic Coulomb and long-range Fröhlich interactions. Here I suggest that a parameter-free evaluation of T_c, unusual upper critical fields, the normal state Nernst effect, diamagnetism, the Hall–Lorenz numbers and giant proximity effects strongly support the three-dimensional (3D) Bose–Einstein condensation (BEC) of mobile small bipolarons with zero off-diagonal order parameter above the resistive critical temperature T_c at variance with phase fluctuation scenarios of cuprates.