  The aim of this thesis is to systematically and consistently study strongly
coupled bosonic and fermionic conformal field theories using the large quantum
number expansion. The idea behind it is to study sectors of conformal field
theories that are characterised by large quantum numbers under global
symmetries. In this limit, the conformal field theories, even if they initially
were strongly coupled and interacting, can now be written in terms of an
effective field theory that is weakly coupled. Some common effective field
theories that appear in the literature are the bosonic conformal superfluid and
the Fermi sphere, condensed matter systems characterised by a high particle
density, making the study of such systems a cross-disciplinary matter.
