Abstract
We briefly review results from extensive quantum Monte Carlo and semiclassical analytic studies of models of ultrasmall Josephson junction arrays (JJA). Specifically, we mention results: (i) close to the renormalized semi-classical critical temperature and its successful comparison to experiment, (ii) The existing renormalization-group, self-consistent harmonic approximation and QMC evidence for a low temperature quantum induced phase transition in a mutual capacitance dominated JJA in zero magnetic field, (iii) duality transformations and phase structure of two capacitively coupled JJA that leads to a structure reminiscent of a fractional quantum Hall effect type system.
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