  We have identified the unscreened Froehlich electron-phonon interaction (EPI)
as the most essential for pairing in cuprate superconductors as now confirmed
by isotope substitution, recent angle-resolved photoemission (ARPES), and some
other experiments. Low-energy physics is that of mobile lattice polarons and
bipolarons in the strong EPI regime. Many experimental observations have been
predicted or explained in the framework of our "Coulomb-Froehlich" model, which
fully takes into account the long-range Coulomb repulsion and the Froehlich
EPI. They include pseudo-gaps, unusual isotope effects and upper critical
fields, the normal state Nernst effect, diamagnetism, the Hall-Lorenz numbers,
and a giant proximity effect (GPE). These experiments along with the
parameter-free estimates of the Fermi energy and the critical temperature
support a genuine Bose-Einstein condensation of real-space lattice bipolarons
in the pseudogap phase of cuprates. On the contrary the phase fluctuation (or
vortex) scenario is incompatible with the insulating-like in-plane resistivity
and the magnetic-field dependence of orbital magnetization in the resistive
state of underdoped cuprates.
