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In the tokamak scrape-off layer (SOL) the magnetic field lines are open, channeling particles and heat onto plasma facing components, and constraining their lifetime. Therefore, safe operation of future fusion reactors requires an understanding of SOL plasma dynamics. Recently, we have gained deep insights into the tokamak SOL dynamics by means of massively-parallel fluid electromagnetic turbulence simulations carried out with the Global Braginskii Solver (GBS) code. GBS is currently capable of performing full-size SOL simulations of medium-size tokamaks such as TCV or Alcator C-Mod. In the present paper, we emphasize recent numerical developments aimed towards (a) more realistic description of the plasma geometry and (b) simulating larger, reactor-class machines. First, we address the numerical implementation of parallel advection and diffusion operators in finite difference representation. This is a crucial aspect of our computation, as the cost of the simulation can be drastically reduced if the parallel dynamics are discretized adequately taking advantage of the strong anisotropy of the turbulent modes that are aligned to the magnetic field lines. Second, we address the development and implementation of a matrix-free, parallel multigrid solver for the Poisson operator in the vorticity equation. Using this new solver, it is now possible to further parallelize GBS without breaking parallel scalability, an important step towards the realm of 10^4 CPUs. Finally, we summarize the understanding of SOL turbulence obtained through GBS simulations - for instance, the mechanisms regulating the turbulence level and therefore the SOL width, the turbulent regimes, and the mechanisms driving plasma rotation in this region. The simulated non-linear dynamics have been compared with analytical estimates, which have highlighted the key physics mechanisms at play in the SOL, and with experimental measurements taken in a number of tokamak worldwide, showing good agreement.