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It is remarkable how the Fröhlich polaron, one of the simplest examples of a Quantum Field Theoretical problem, as it basically consists of a single fermion interacting with a scalar Bose field of ion displacements, has resisted full analytical or numerical solution at all coupling since ∼ 1950, when its Hamiltonian was first written. The field has been a testing ground for analytical, semi-analytical, and numerical techniques, such as path integrals, strong-coupling perturbation expansion, advanced variational, exact diagonalisation (ED), and quantum Monte Carlo (QMC) techniques. This article reviews recent developments in the field of continuum and discrete (lattice) Fröhlich (bi)polarons starting with the basics and covering a number of active directions of research. * Electronic address: jozef.devreese@ua.ac.be † Electronic address: a.s.alexandrov@lboro.ac.uk

Cooperative properties of self-localized caniers on a lattice are discussed and reviewed at a fairly basic level with an 'emphasis on developments of the strong-coupling theory of superconductivity for the interpretation of high-T, superconductors. Small polaron and bipolaron formation is shown to provide a number of new physical phenomena both in the normal and superconducting states. Two mechanisms of superconductivity are discussed in detail. The first one arises from the Cooper pairing of small polarons in momentum space (polaronic superconductivity) while the second one is due to polaron pairing in real space and analogous to the superfluidity of dHe (bipolaronic superconductivity). Highly non-adiabatic motion of bipolarons results in fundamental differences of bipolaronic superconductivity with respect to the BCS one including its well known strong-coupling generalization. The review covers the theoretical development and some experimental results in the past decade paying special attention to the physical properties of high-T, oxides and their explanation with (bi)polarons. Basic properties of charged bosons are also considered.

Extending the Fröhlich polaron problem to a discrete ionic lattice we study a polaronic state with a small radius of the wave function but a large size of the lattice distortion. We calculate the energy dispersion and the effective mass of the polaron with the 1/λ perturbation theory and with the exact Monte Carlo method in the nonadiabatic and adiabatic regimes, respectively. The "small" Fröhlich polaron is found to be lighter than the small Holstein polaron by one or more orders of magnitude. 71.38.+i,74.20.Mn

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