Heavy fermions form a prominent class of strongly-correlated electron systems. Central to these compounds is a lattice of local moments, coupled to each other and to conduction electrons. The interplay between the two types of interactions gives rise to magnetic, quantum phase transitions. On the paramagnetic side, Kondo-screening ensures that the moments are absorbed into an emerging Fermi-liquid. On the ordered side, the moments decouple from the conduction electrons, leading to the suppression of Kondo-screening. These general considerations motivate local quantum criticality, which is associated with a zero temperature antiferromagnetic phase transition involving the destruction of a Kondo effect, and the associated Global Phase Diagram. In the first part of this work, I present a study of the Kondo-destruction criticality. I examined the stability of locally-critical transitions with respect to the additional effects of quantum fluctuations, inherent to the lattice of magnetic moments. The method was provided by the Extended Dynamical Mean-Field Theory, which solves an impurity model, self-consistently embedded in the background of the critical lattice. The numerical and analytical results indicate that locally-critical transitions are stable with respect to enhanced quantum fluctuations, providing an important validation of the Global Phase Diagram. The work also includes a preliminary discussion of Kondo-destruction criticality in systems with additional quadrupole localized degrees-of-freedom.

The second part is dedicated to the study of unconventional superconductivity in Fe-based compounds. These inherently multi-orbital systems allow for the possibility of exploring novel pairing states with non-trivial orbital structure. I argue that a unconventional pairing function, dubbed s⊗τ3, likely explains the experimental results for the Fe-selenide class, which would otherwise be very difficult to reconcile through the more common, simple s- and d-wave pairing. Using a two-orbital toy model, I illustrate how such an orbitally-entangled pairing state can simultaneously exhibit properties typically associated with either s- or d-wave pairing functions. I further present detailed numerical calculations for a realistic five-orbital model, which show that the orbitally-entangled $ s \otimes \tau_{3}$ state can be stabilized for a range of tuning parameters. Brief discussions of various non-trivial aspects of symmetry and of the extension to other similar superconductors are given.