  We study correlated quantum impurity models which undergo a local quantum
phase transition (QPT) from a strong coupling, Fermi liquid phase to a
non-Fermi liquid phase with a globally doubly degenerate ground state. Our aim
is to establish what can be shown exactly about such `local moment' (LM)
phases; of which the permanent (zero-field) local magnetization is a hallmark,
and an order parameter for the QPT. A description of the zero-field LM phase is
shown to require two distinct self-energies, which reflect the broken symmetry
nature of the phase and together determine the single self-energy of standard
field theory. Distinct Friedel sum rules for each phase are obtained, via a
Luttinger theorem embodied in the vanishing of appropriate Luttinger integrals.
By contrast, the standard Luttinger integral is non-zero in the LM phase, but
found to have universal magnitude. A range of spin susceptibilites are also
considered; including that corresponding to the local order parameter, whose
exact form is shown to be RPA-like, and to diverge as the QPT is approached.
Particular attention is given to the pseudogap Anderson model, including the
basic physical picture of the transition, the low-energy behavior of
single-particle dynamics, the quantum critical point itself, and the rather
subtle effect of an applied local field. A two-level impurity model which
undergoes a QPT (`singlet-triplet') to an underscreened LM phase is also
considered, for which we derive on general grounds some key results for the
zero-bias conductance in both phases.

Strongly Correlated Electrons