The emergence of metalenses has impacted a wide variety of applications such as beam steering, imaging, depth sensing, and display projection. Optical distortion, an important metric among many optical design specifications, has however rarely been discussed in the context of meta-optics. Here, we present a generic approach for on-demand distortion engineering using compound metalenses. We show that the extra degrees of freedom afforded by a doublet metasurface architecture allow custom-tailored angle-dependent image height relations and hence distortion control while minimizing other monochromatic aberrations. Using this platform, we experimentally demonstrate a compound fisheye metalens with diffraction-limited performance across a wide field of view of 140° and a low barrel distortion of less than 2%, compared with up to 22% distortion in a reference metalens without compensation. The design strategy and compound metalens architecture presented herein are expected to broadly impact metasurface applications in consumer electronics, automotive and robotic sensing, medical imaging, and machine vision systems.