  Modified Maxwell electrodynamics, or ModMax for short, is the unique
nonlinear extension of Maxwell's theory that preserves its notable symmetries:
conformal invariance and electromagnetic duality. ModMax has been studied
extensively at the classical level, however remains largely untouched in a
quantum context due to its non-analytic nature.
  In this thesis, we perform the perturbative quantization of this theory.
Using the background field method and dimensional regularization, we obtain
novel corrections by calculating the one loop quantum effective action. These
corrections vanish in a background with constant field strength, and are not of
the form of the classical theory for a general background field. Motivated by
the corrections obtained for ModMax, we applied the method developed to
quantize ModMax to its two dimensional analogue theory. We similarly obtain the
one loop quantum effective action for this theory in a general background by
evaluating all one loop Feynman diagrams. In addition, we study the divergence
of the separate infinite series of two vertex diagrams.
