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In the tokamak scrape-off layer (SOL), turbulent plasma interacts with the wall, determining the boundary conditions for the core plasma, and largely governing the performance of the entire device. In this region, the amplitude of the turbulent modes is comparable to the background plasma profiles, and there is no separation between turbulent and equilibrium length scales. Therefore, a fully non-linear and global approach is necessary. The present work discusses recent studies addressing the properties of tokamak SOL turbulence using a global, electromagnetic, fluid drift-reduced Braginskii model. Three-dimensional, non-linear simulations are carried out using the Global Braginskii Solver (GBS) code [1], which is now capable of carrying out self-consistent, global three-dimensional simulations of the plasma dynamics in the tokamak SOL. The simulations evolve the plasma dynamics as the interplay of the plasma flux from the core, the turbulent radial transport, and the losses at the plasma sheath where the magnetic field lines intersect with the vessel. A gradual approach in increasing complexity has made possible (a) to determine the dominant instabilities driving the SOL turbulence, (b) to identify the mechanisms that saturate the growth of the linear modes and therefore regulate level of radial transport, (c) to characterize the plasma intrinsic rotation induced by the SOL dynamics, and (d) to study the role of electromagnetic effects in enhanced transport regimes. The non-linear dynamics revealed by the simulations agree with the analytical estimates that have been carried out. [1] P.Ricci et al., Plasma Physics and Controlled Fusion, 2012, 54, 124047.