This thesis is devoted to the study of the incompressible and stationary Navier–Stokes equations in two-dimensional unbounded domains. In 1933, Leray proved the existence of weak solutions for the stationary Navier–Stokes equations in two and three dimensions. In three dimensions the function spaces used by Leray allow to characterize the limit of the velocity field at infinity when the domain is unbounded. However, in two dimensions, the function spaces in which the weak solutions are constructed do not allow to obtain any information on the behavior of the velocity at infinity for unbounded domains. Despite the efforts of many mathematicians, the existence of solutions whose velocity field converges to a constant vector at infinity is still an open problem in two dimensions. When the data are small, Finn & Smith proved in 1967 the existence of solutions whose velocity field converges to any prescribed vector field that is nonzero. However, the existence of solutions whose velocity field goes to zero at infinity is an open problem, even for small data. This thesis is mainly devoted to the study of this latter case, from an analytical and numerical point of view.