  We review the Extended Mean Field (EMFT) approximation and apply it to
complex, scalar $\varphi^4$ theory on the lattice. We determine the ($T,\mu$)
phase diagram and study the critical properties of the Bose condensation
transition, at zero and finite temperature. In both cases we obtain results
which agree very well with recent Monte Carlo studies. Within our approximation
we can reach the thermodynamic limit and can thus obtain results at lattice
spacings unreachable by present Monte Carlo simulations. We find that our
approximation reproduces accurately many phenomena of the model, like the
"Silver Blaze" behavior at zero temperature and dimensional reduction at finite
temperature.
