  We sketch recent interactions between model theory and a roughly 150-year old
study of analytic functions involving complex analysis, algebraic topology, and
number theory, centered in canonicity of universal covers. Towards this goal we
discuss in a systematic and unified way several examples indicating the main
ideas of the proofs and the necessary changes in method for different
situations: exponential covers, modular and Shimura curves, Shimura and abelian
varieties, and coherent families of smooth covers.
