  A comprehensive analysis of half-doped manganites is presented using Monte
Carlo simulations applied to the double-exchange model with cooperative
Jahn-Teller lattice distortions in two dimensions. A variety of novel results
are reported: (i) The phase diagram is established in the lambda-J_ AF plane,
with lambda the electron-phonon coupling and J_AF the antiferromagnetic
exchange between classical t_2g spins. The results include standard phases,
such as the CE-insulating and FM-metallic regimes, but they also unveil novel
states, such as a ferromagnetic charge-ordered (CO) orbital-ordered phase
compatible with recent experimental results by Loudon et al. (ii) For realistic
couplings, it was observed that the charge disproportionation delta of the CO
phase is far from the widely accepted extreme limit delta=0.5 of a 3+/4+ charge
separation. A far smaller delta appears more realistic, in agreement with
recent experiments by Garcia et al. and Daoud-Aladine it et al. (iii) Colossal
magnetoresistance (CMR) effects are found in calculations of cluster
resistances using the Landauer formalism. This occurs near the ubiquitous
first-order phase transitions between the insulating and metallic states. (iv)
The CE-state is found to be (very sensitive to disorder) since its long-range
order rapidly disappears when quenched-disorder is introduced, contrary to the
FM state which is more robust. This is also in qualitative agreement with
recent experiments by Akahoshi et al. and Nakajima et al. (v) The phase diagram
in the half-doped (electron doping) regime is briefly discussed as well. A
charge-ordered state is found which is the analog of the x=0.5 CE phase. It
contains a 3+/2+ charge arrangement at large lambda.
