Quantum field theories of self-interacting fermions play an important
role in particle and condensed matter physics, covering effective
descriptions of the strong nuclear force, the critical behaviour of
Dirac materials such as graphene, and more. In this talk, I discuss
fermionic systems in the large-N limit using the method functional
renormalisation. Working directly in terms of fermionic field variables,
I provide conditions under which fermionic functional RG flows become
exact, and exactly solvable, and give the general form of their quantum
effective actions. I exemplify the method for fermionic theories with
scalar, pseudo-scalar, vector, axial-vector, and derivative interactions
in various dimensions, and use it to study non-perturbative phenomena
such as chiral symmetry breaking and dynamical generation of fermion
mass, interacting fixed points and universal operator scaling
dimensions, as well as features such as conformal manifolds with exactly
marginal interactions, the spontaneous breaking of scale symmetry, and
the appearance of a massless dilaton. Exact dualities with bosonised
versions of theories are also discussed.