Since its introduction in 1963, the Hubbard model has becomes one of the most popular models used in the literature to study cooperative phenomena in narrow-band metals (ferromagnetism, metal-insulator transitions, charge-density waves, high-Tc superconductivity). Amongst all these cooperative phenomena, the problem of itinerant ferromagnetism in the Hubbard model has the longest history. However, in spite of an impressive research activity in the past, the underlying physics (microscopic mechanisms) that leads to the stabilization of itinerant ferromagnetism in Hubbard model (narrow-band metals) is still far from being understood. In this review we present our numerical results concerning this subject, which have been reached by small cluster exact diagonalization, density matrix renormalization group and quantum Monte Carlo calculations within various extensions of the Hubbard model. Particular attention is paid to a description of crucial mechanisms (interactions) that support the stabilization of the ferromagnetic state, and namely: (i) the long-range hopping, (ii) the correlated hopping, (iii) the long-range Coulomb interaction, (iv) the flat bands and (v) the lattice structure. Most of the presented results have been obtained for the one-dimensional case, but the influence of the increasing dimension of the system on the ferromagnetic state is also intensively discussed.
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