  The amplification of the primordial magnetic fields and the gravitational
baryogenesis, a mechanism that allows to generate the baryon asymmetry in the
Universe by means of the coupling between the Ricci scalar curvature and the
baryon current, are reviewed in the framework of the nonlinear electrodynamics.
To study the amplification of the primordial magnetic field strength, we write
down the gauge invariant wave equations and then solve them (in the long
wavelength approximation) for three different eras of the Universe: de Sitter,
the reheating and the radiation dominated era. Constraints on parameters
entering the nonlinear electrodynamics are obtained by using the amplitude of
the observed galactic magnetic fields and the baryon asymmetry, which are
characterized by the dimensionless parameters $r\sim 10^{-37}$ and
$\eta_B\lesssim 9\times 10^{-11}$, respectively.

Cosmology and Nongalactic Astrophysics