This thesis deals with the stochastic Navier-Stokes equations in both the incompressible and the compressible case. Our goal in both situations is, roughly speaking, to present a denition of solutions to the SPDE systems and afterwards to prove existence of those solutions. The second part of the thesis considers the incompressible Navier-Stokes equations. It is shown that the incompressible Navier-Stokes equations admit a formulation as a stochastic evolution equation in the space of divergence free vector fields.The concept of martingale solutions for stochastic evolution equations is introduced and the existence martingale solutions as well as the existence of almost sure martingale solutions to the Markov problem is prooved. In the third part of the present thesis, the compressible Navier-Stokes equations are studied. The extension of the concept of finite energy weak solutions to the stochastic case is discussed. The heart of the third part is the proof of existence of finite energy weak solutions to the compressible Navier-Stokes system governed by distributional forces. The proof of existence of solutions to the stochastic Navier-Stokes system makes use of a measurable selection theorem. In the final section of this thesis, the developed theory is applied to Levy processes.