We study correlated two-level quantum impurity models coupled to a metallic con- duction band in the hope of gaining insight into the physics of nanoscale quantum dot systems. We focus on the possibility of formation of a spin-I impurity local moment which, on coupling to the band, generates an underscreened (USC) singular Fermi liq- uid state. By employing physical arguments and the numerical renorrnalization group (NRG) technique, we analyse such systems in detail examining in particular both the thermodynamic and dynamic properties, including the differential conductance. The quantum phase transitions occurring between the USC phase and a more ordinary Fermi liquid (FL) phase are analysed in detail. They are generically found to be of Kosterlitz- Thouless type; exceptions occur along lines of high symmetry where first-order transitions are found. A 'Friedel-Luttinger sum rule' is derived and, together with a generalization of Luttinger's theorem to the USC phase, is used to obtain general results for the T = 0 zero-bias conductance ~ it is expressed solely in terms of the number of electrons present on the impurity and applicable in both the USC and FL phases. Relatedly, dynamical signatures of the quantum phase transition show two broad classes of behaviour corresponding to the collapse of either a resonance or antiresonance in the single-particle density of states. Evidence of both of these behaviours is seen in experimental devices. We study also the effect of a local magnetic field on both single- and two-level quantum impurities. In the former case we attempt -to resolve some points of con- tention that remain in the literature. Specifically we show that the position of the maximum in the spin resolved density of states (and related peaks in the diffcrcn- tial conductance) is not linear in the applied field, showing a more complicated form than a simple 'Zeeman splitting'. The analytic result for the low-field asymptote is recovered. For two-level impurities we illustrate the manner in which the USC sta.te is destroyed: due to two cancelling effects an abrupt change in the zero-bias conductance does not occur as one might expect. Comparison with experiment is made in both cases and used to interpret experimental findings in a manner contrary to previous suggestions. We find that experiments are very rarely in the limit of strong impurity- host coupling. Further, features in the differential conductance as a function of bias voltage should not be simply interpreted in terms of isolated quantum dot states. The many-body nature of such systems is crucially important to their observed properties.