  Josephson junction arrays are ideal model systems where a variety of
phenomena, phase transitions, frustration effects, vortex dynamics, chaos, to
mention a few of them, can be studied in a controlled way. In this review we
focus on the quantum dynamical properties of low capacitance Josephson junction
arrays. The two characteristic energy scales in these systems are the Josephson
energy, associated to the tunneling of Cooper pairs between neighboring
islands, and the charging energy, which is the energy cost to add an extra
electron charge to a neutral island. The phenomena described in this review
stem from the competition between single electron effects with the Josephson
effect. One example is the (quantum) Superconductor-Insulator phase transition
which occurs by varying the ratio between the coupling constants and/or by
means of external magnetic/electric fields. We will describe how the phase
diagram depends on the various control paramters and the transport properties
close to the quantum critical point. The relevant topological excitations on
the superconducting side of the phase diagram are vortices. In low capacitance
junction arrays vortices behave as massive underdamped particles that can
exhibit quantum behaviour. We will report on the various experiments and
theoretical treatments on quantum vortex dynamics.
