As is well known, one of the most studied questions in theoretical and applied fluid mechanics is the steady-state, plane, exterior boundary-value problem associated to the Navier-Stokes equations. The problem, in its mathematical formulation, consists in finding a vector function v = (v1, v2) and a scalar function p, depending only on x = (x1, x2) that satisfy the following system of equations ∆v= λv∙∇v+ ∇p in Ω ∇v=0 in Ω (1) v|∂Ω=v_(* ),lim┬(|x|→∞)⁡v(x)= ξ. In (1), Ω⊂R^2 is the exterior of a two-dimensional compact, connected set of R^2, ξ = (ξ1, ξ2 ) is a fixed constant vector, v* = (v*1, v*2) is a prescribed vector function at the boundary ∂Ω of Ω , and λ is a given non-negative real number (Reynolds number). From the physical point of view, the system of equations (1) describes the stationary motion of a viscous, incompressible fluid around a long, straight cylinder C, assuming that the fluid flow is uniform at large distances from C. Despite its great relevance, and the attempts by many mathematicians, the problem presents several fundamental open questions. The most significant is, undoubtedly, the existence of a solution for arbitrarily large Reynolds number (“laminar” solution). As a matter of fact, it is not even clear whether or not such a problem admits a positive answer. In this respect, neither the experiments nor a numerical computation (so far) can be of help, in that a steady-state solution becomes unstable at very low Reynolds number (~ 40). The objective of this talk is to give a complete and updated overview of the main known results about the existence problem, and to provide some possible ways for proving (or disproving) its resolution. In particular, it will be shown that the lack of existence of solutions for arbitrary λ would have consequences that are very questionable on physical ground, and that would give less credibility to the Navier-Stokes model. An effort will be made to avoid technical details, with the objective of making the presentation accessible also to non-specialists.