  In the quest to produce quantum technology, superconducting networks, working
at temperatures just above absolute zero, have arisen as one of the most
promising physical implementations. The precise analysis and synthesis of such
circuits have required merging the fields of physics, engineering, and
mathematics.
  In this dissertation, we develop mathematically consistent and precise
Hamiltonian models to describe ideal superconducting networks made of an
arbitrary number of lumped elements, such as capacitors, inductors, Josephson
and phase-slip junctions, gyrators, etc., and distributed ones like
transmission lines. We give formal proofs for the decoupling at high and low
frequencies of lumped degrees of freedom from infinite-dimensional systems in
different coupling configurations in models based on the effective Kirchhoff's
laws. We extend the standard theory to quantize circuits that include ideal
nonreciprocal elements all the way to their Hamiltonian descriptions in a
systematic way. Finally, we pave the way on how to quantize general
frequency-dependent gyrators and circulators coupled to both transmission lines
and other lumped-element networks.
  We have explicitly shown, that these models, albeit ideal, are finite and
present no divergence issues. We explain and dispel misunderstandings from the
previous literature. Furthermore, we have demonstrated the usefulness of a
redundant basis for performing separation of variables of the transmission line
(1D) fields in the presence of point-like (lumped-element) couplings by
time-reversal symmetry-breaking terms, i.e. nonreciprocal elements.
