These notes are based on a series of lectures delivered by the author at the University of Toulouse in February 2014. They are entirely devoted to the initial value problem and the long-time behavior of solutions for the two-dimensional incompressible Navier-Stokes equations, in the particular case where the domain occupied by the fluid is the whole plane R 2 and the velocity field is only assumed to be bounded. In this context, local well-posedness is not difficult to establish [19], and a priori estimates on the vorticity distribution imply that all solutions are global and grow at most exponentially in time [20, 40]. Moreover, as was recently shown by S. Zelik, localized energy estimates can be used to obtain a much better control on the uniformly local energy norm of the velocity field [46]. The aim of these notes is to present, in an explanatory and self-contained way, a simplified and optimized version of Zelik's argument which, in combination with a new formulation of the Biot-Savart law for bounded vorticities, allows one to show that the L ∞ norm of the velocity field grows at most linearly in time. The results do not rely on the viscous dissipation, and remain therefore valid for the so-called "Serfati solutions" of the two-dimensional Euler equations [2]. In the viscous case, a recent work by S. Slijepčevi´Slijepčevi´c and the author shows that all solutions stay uniformly bounded if the velocity field and the pressure are periodic in a given space direction [16, 17].