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The first direct detection of gravitational waves from the mergers of binary black holes and binary neutron stars by LIGO/Virgo missions has electrified the physics and astronomy communities. An even more intriguing signal is the stochastic gravitational wave background produced by turbulent plasma motions in the early-universe. I will discuss the stochastic gravitational wave background generated in the early universe by plasma turbulence and resulting magnetic fields. The possibility of turbulent motions presence in the early universe is also crucial when addressing cosmic magnetism origin. In fact, the origin of cosmic magnetic fields is one of the big open questions in astrophysics and space physics. It is generally thought that these magnetic fields are the result of the amplification of weak initial seed fields. Lower limits on extragalactic magnetic fields from blazar observations suggest seed magnetic fields generated in the early universe. I will address our numerical simulations of primordial plasma turbulent motions and magnetic field amplification using the PENCIL magnetohydrodynamic (MHD) code. I will discuss the primordial magnetic fields effects on different cosmological observables (such as cosmic microwave radiation, large scale structure formation, etc), and correspondingly I will address the magnetic field limits from currently available observational data. Finally I will present our results of the gravitational wave signals resulting from both the plasma motions and magnetic fields.