  In this work, we introduce and study the well-posedness of the
multidimensional fractional stochastic Navier-Stokes equations on bounded
domains and on the torus (Briefly dD-FSNSE). We prove the existence of a
martingale solution for the general regime. We establish the uniqueness in the
case a martingale solution enjoys a condition of Serrin's type on the
fractional Sobolev spaces. If an $L^2-$ local weak (strong in probability)
solution exists and enjoys conditions of Beale-Kato-Majda type, this solution
is global and unique. These conditions are automatically satisfied for the
2D-FSNSE on the torus if the initial data has $ H^1-$regularity and the
diffusion term satisfies growth and Lipschitz conditions corresponding to $
H^1-$spaces. The case of 2D-FSNSE on the torus is studied separately. In
particular, we established thresholds for the global existence, uniqueness,
space and time regularities of the weak (strong in probability) solutions in
the subcritical regime.
