We formulate and solve the inverse problem of quartic photonics, or photonics of metamaterials, whose k-surfaces (also known as iso-frequency surfaces or Fresnel surfaces) are quartic surfaces. This provides an approach to engineer the effective parameters of metamaterials starting from the desired plane waves. We apply our method to the design of the high-k limit of metamaterials, extreme non-reciprocity, and complex bi-anisotropic media.