After decades of being mostly confined to the local Universe, the study of gas dynamics has now become an essential tool to follow the evolution of galaxies across cosmic time. Gas rotation in galaxies allows us to trace the distribution of matter, quantify the mass of the dark matter halos and study galaxy scaling relations. In contrast, gas turbulence reveals the effects of stellar feedback and disc instabilities and provides clues about the formation of the stellar thin and thick discs. Moreover, rotation measurements allow us to estimate the build up of angular momentum in galaxies, which provides key tests for theoretical models of galaxy formation. In this talk, I will present results on high-z rotation curves and velocity dispersions, obtained through 3D reconstruction techniques of the emission-line datacubes. After discussing results at z~1, I will focus on ALMA observations of starburst galaxies at z~4 that reveal very fast rotation and surprisingly low velocity dispersions leading to V/sigma values >10. These results, on the one hand, provide dynamical evidence of the evolution of these massive systems into local early-type galaxies, but on the other hand, pose new critical challenges to our current understanding of galaxy formation at early times.