In this paper, we review the generation of vortex beams carrying orbital angular momentum in the microwave domain. We firstly present the theory of Laguerre–Gaussian beams where it is demonstrated that they carry such type of momentum. We further provide an overview of the classical methods used to generate orbital angular momentum vortex beams, which rely on two main methods; plane wave to vortex wave conversion and direct generation using radiating antennas. Then, we present recent progress in the physics of metasurfaces devoted to the generation of vortex beams with a discussion about reflective and transmissive metasurfaces for plane wave to vortex wave conversion as well as methods to reduce the intrinsic divergence characteristics of vortex beams. Finally, we conclude on this rapidly developing research field.