  We calculate the superfluid transition temperature of homogeneous interacting
Bose gases in three and two spatial dimensions using large-scale Path Integral
Monte Carlo simulations (with up to $N=10^5$ particles). In 3D we investigate
the limits of the universal critical behavior in terms of the scattering length
alone by using different models for the interatomic potential. We find that
this type of universality sets in at small values of the gas parameter $na^3
\lesssim 10^{-4}$. This value is different from the estimate $na^3 \lesssim
10^{-6}$ for the validity of the asymptotic expansion in the limit of vanishing
$na^3$. In 2D we study the Berezinskii-Kosterlitz-Thouless transition of a gas
with hard-core interactions. For this system we find good agreement with the
classical lattice $|\psi|^4$ model up to very large densities. We also explain
the origin of the existing discrepancy between previous studies of the same
problem.
