  The phase diagram of two dimensional Josephson arrays is studied by means of
the mapping to the quantum XY model. The quantum effects onto the
thermodynamics of the system can be evaluated with quantitative accuracy by a
semiclassical method, the {\em pure-quantum self-consistent harmonic
approximation}, and those of dissipation can be included in the same framework
by the Caldeira-Leggett model. Within this scheme, the critical temperature of
the superconductor-to-insulator transition, which is a
Berezinskii-Kosterlitz-Thouless one, can be calculated in an extremely easy way
as a function of the quantum coupling and of the dissipation mechanism.
Previous quantum Monte Carlo results for the same model appear to be rather
inaccurate, while the comparison with experimental data leads to conclude that
the commonly assumed model is not suitable to describe in detail the real
system.
