  Theories of non-linear electrodynamics naturally describe deviations from
Maxwell's theory in the strong field regime. Among these, of special interest
is the recently discovered ModMax electrodynamics, which is a unique
1-parametric generalization of Maxwell's theory that possesses the conformal
invariance as well as the electromagnetic duality. In this paper we construct
the asymptotically AdS accelerated black holes in this theory and study their
thermodynamics - providing thus a first example of accelerated solutions
coupled to non-linear electrodynamics. Our study opens a window towards
studying radiative spacetimes in non-linear electromagnetic regime as well as
raises new challenges for their corresponding holographic interpretation.
