We show how the deposition of laser energy induces propulsion and strong deformation of an absorbing liquid body. Combining high speed with stroboscopic imaging, we observe that a millimeter-sized dyed water drop hit by a millijoule nanosecond laser pulse propels forward at several meters per second and deforms until it eventually fragments. The drop motion results from the recoil momentum imparted at the drop surface by water vaporization. We measure the propulsion speed and the time-deformation law of the drop, complemented by boundary-integral simulations. The drop propulsion and shaping are explained in terms of the laser-pulse energy, the drop size, and the liquid properties. These findings are, for instance, crucial for the generation of extreme ultraviolet light in nanolithography machines.