Terrestrial planet formation is believed to have concluded in our Solar System with about 10million to 100million years of giant impacts, where hundreds of Moon- to Mars-sized planetary embryos acquired random velocities through gravitational encounters and resonances with one another and with Jupiter. This led to planet-crossing orbits and collisions that produced the four terrestrial planets, the Moon and asteroids. But here we show that colliding planets do not simply merge, as is commonly assumed. In many cases, the smaller planet escapes from the collision highly deformed, spun up, depressurized from equilibrium, stripped of its outer layers, and sometimes pulled apart into a chain of diverse objects. Remnants of these `hit-and-run' collisions are predicted to be common among remnant planet-forming populations, and thus to be relevant to asteroid formation and meteorite petrogenesis.