Precision Weak Lensing: How to measure what cannot be seen

In our current view of the Universe, galaxies form and evolve within dark matter halos. However, understanding how the invisible dark matter halos relate to galaxies and which galaxy properties are affected by them remains a complex task. This thesis explores a new methodology to measure the mass of dark matter haloes by studying lensing-induced deformations in the observed velocity fields of lensed galaxies. Overall, we demonstrate that the new approach provides ten-times more accurate measurements than previous methods, and allows to explore new important quantities in the formation and evolution of galaxies.