Rotation is one of the important parameters affecting the evolution and final fate of massive stars but the origin of fast rotators remains unclear (imprint of the star formation process, result of binary interactions). In this work, we aim at investigating the binary status, photometric variability, and runaway status of a statistically meaningful sample of Galactic fast-rotating O stars. We perform a comprehensive multi-epoch analysis of new high-quality spectroscopic observations gathered by the IACOB and OWN surveys. Notably, we find that the total percentage of spectroscopic binaries in the investigated sample range between 25 and 40%, in agreement with previous finding for the case of O-type stars with lower projected rotational velocities. On the contrary, the fraction of runaway stars among fast rotators (∼35-50%) is significantly higher than in the case of slow rotators (∼20-30%). By combining all these observational results we will evaluate each scenario about the origin of fast rotators.