  We study the effects of external offset charges on the phase diagram of
Josephson junction arrays. Using the path integral approach, we provide a
pedagogical derivation of the equation for the phase boundary line between the
insulating and the superconducting phase within the mean-field theory
approximation. For a uniform offset charge q=e the superconducting phase
increases with respect to q=0 and a characteristic lobe structure appears in
the phase diagram when the critical line is plotted as a function of q at fixed
temperature. We review our analysis of the physically relevant situation where
a Josephson network feels the effect of random offset charges. We observe that
the Mott-insulating lobe structure of the phase diagram disappears for large
variance (\sigma > e) of the offset charges probability distribution; with
nearest-neighbor interactions, the insulating lobe around q=e is destroyed even
for small values of \sigma. Finally, we study the case of random
self-capacitances: here we observe that, until the variance of the distribution
reaches a critical value, the superconducting phase increases in comparison to
the situation in which all self-capacitances are equal.
