Uniform triangular crystals are the ground state for particles that interact isotropically in two dimensions. However, when immersed in an external potential, for example, one arising from an electric field, a flow field, or gravity, the resulting phases are significantly distorted in a way reminiscent of conformal transformations of planar lattices. We study these "conformal crystals" using colloidal experiments and molecular dynamics simulations. By establishing a projection from these self-assembled inhomogeneous crystals to homogeneous crystals on curved surfaces, we are able to both predict the distribution of defects and establish that defects are an almost inevitable part of the ground state. We determine how the inherent geometry emerges from an interplay between the confining potential and the interparticle interactions. Using molecular dynamics simulations, we demonstrate the generic behavior of this emergent geometry and the resulting defect structures throughout a variety of physical systems.