  We investigate the Bose glass phase and the insulator-to-superfluid
transition in the two-dimensional disordered boson Hubbard model in the Villain
representation via Monte Carlo simulations. In the Bose glass phase the
probability distribution of the local susceptibility is found to have a $1/
\chi^2$ tail and the imaginary time Green's function decays algebraically
$C(\tau) \sim \tau^{-1}$, giving rise to a divergent global susceptibility. By
considering the participation ratio it is shown that the excitations in the
Bose glass phase are fully localized and a scaling law is established. For
commensurate boson densities we find a direct Mott insulator to superfluid
transition without an intervening Bose glass phase for weak disorder. For this
transition we obtain the critical exponents $z=1, \nu=0.7\pm 0.1$ and $\eta =
0.1 \pm 0.1$, which agree with those for the classical three-dimensional XY
model without disorder. This indicates that disorder is irrelevant at the tip
of the Mott-lobes and that here the inequality $\nu\ge2/d$ is violated.
