  The phase diagram of the ferromagnetic Kondo model for manganites is
investigated using computational techniques. In clusters of dimensions 1 and 2,
Monte Carlo simulations in the limit where the localized spins are classical
show a rich low temperature phase diagram with three dominant regions: (i) a
ferromagnetic phase, (ii) phase separation between hole-poor antiferromagnetic
and hole-rich ferromagnetic domains, and (iii) a phase with incommensurate spin
correlations. Possible experimental consequences of the regime of phase
separation are discussed. Studies using the Lanczos algorithm and the Density
Matrix Renormalization Group method applied to chains with localized spin 1/2
(with and without Coulombic repulsion for the mobile electrons) and spin 3/2
degrees of freedom give results in excellent agreement with those in the spin
localized classical limit. The Dynamical Mean Field ($D=\infty$) approximation
was also applied to the same model. At large Hund coupling phase separation and
ferromagnetism were identified, again in good agreement with results in low
dimensions. In addition, a Monte Carlo study of spin correlations allowed us to
estimate the critical temperature for ferromagnetism $T_c^{FM}$ in 3
dimensional clusters. It is concluded that $T_c^{FM}$ is compatible with
current experimental results.
