  In this thesis, we perform a comprehensive renormalization group analysis of
two- and three-dimensional Fermi systems at low and zero temperature. We
examine systems with spontaneous symmetry-breaking and quantum critical
behavior by deriving and solving flow equations within the functional
renormalization group framework. We extend the Hertz-Millis theory of quantum
phase transitions in itinerant fermion systems to phases with discrete and
continuous symmetry-breaking, and to quantum critical points where the zero
temperature theory is associated with a non-Gaussian fixed point. We compute
the finite temperature phase boundary near the quantum critical point
explicitly including non-Gaussian fluctuations. We then set up a coupled
fermion-boson renormalization group theory that captures the mutual interplay
of gapless fermions with massless order parameter fluctuations when approaching
a quantum critical point. As a first application, we compute the complete set
of quantum critical exponents at the semimetal-to-superfluid quantum phase
transition of attractively interacting Dirac fermions in two dimensions. Both,
the order parameter propagator and the fermion propagator become non-analytic
functions of momenta destroying the Fermi liquid behavior. We finally compute
the effects of quantum fluctuations in the superfluid ground state of an
attractively interacting Fermi system, employing the attractive Hubbard model
as a prototype. The flow equations capture the influence of longitudinal and
Goldstone order parameter fluctuations on non-universal quantities such as the
fermionic gap and the fermion-boson vertex, as well as the exact universal
infrared asymptotics present in every fermionic superfluid.
