  Over the last decade several competing models of high-temperature
superconductivity were proposed, most of them with short-range interactions. We
review a more realistic model with strong on-site repulsive correlations, the
Coulomb and strong finite-range electron-phonon interactions. Bipolarons in the
model exist in the itinerant Bloch states at temperatures below about half of
the characteristic phonon frequency. Depending on the ratio of the inter-site
Coulomb repulsion and the polaron level shift the ground state of the model is
a polaronic Fermi (or Luttinger) liquid, bipolaronic high-Tc superconductor, or
charge-segregated insulator for the strong, intermediate, and weak Coulomb
repulsion, respectively. Two particular lattices are analysed in detail: a
chain with the finite range electron-phonon interaction and a zig-zag ladder.
Charge carriers in the ladder are superlight mobile intersite bipolarons. They
propagate coherently without emission or absorption of phonons with about the
same mass as single polarons. The model describes key features of the cuprates,
in particular their Tc values, different isotope effects, normal state
pseudogaps, and spectral functions measured in tunnelling and photoemission.
