The aim of this talk is to discuss the existence of stationary weak solutions to the Navier–Stokes equations in the whole space $\mathbb{R}^3$ and mainly in the whole plane $\mathbb{R}^2$. Contrary to the Cauchy problem for which the $\mathbb{R}^3$ domain is more complicated than $\mathbb{R}^2$, for the stationary problem, this is the opposite: $\mathbb{R}^2$ is the most difficult case. I will discuss the difficulties and how the invading method of Leray can be adapted to construct weak solutions in $\mathbb{R}^2$ including their asymptotic behavior. Joint work with Mikhail Korobkov, Xiao Ren, and Peter Wittwer.