  We review various cosmological models with a local underdense region (local
void) and the averaged models with the backreaction of inhomogeneities, which
have been proposed to explain (without assuming a positive cosmological
constant) the observed accelerating behaviors appearing in the
magnitude-redshift relation of SNIa. To clarify their reality, we consider
their consistency with the other observational studies such as CMB temperature
anisotropy, baryon acoustic oscillation, kinematic Sunyaev-Zeldovich effect,
and so on. It is found as a result that many inhomogeneous models seem to be
ruled out and only models with the parametrs in the narrow range remain to be
examined, and that, unless we assume very high amplitudes of perturbations or
gravitational energies, the averaged models cannot have the accelerated
expansion and the fitted effective Lambda has not the value necessary for the
observed acceleration.
