In this talk, I will discuss recent advances in the computability of various objects arising in Complex Dynamics and the Theory of Univalent functions. After a brief introduction to the general Computability Theory, I will talk about the computation properties of polynomial Julia sets, conformal maps, and Harmonic measure. Based on joint work with M. Braverman (Princeton), A. Glucksam (Northwestern), C. Rojas ( Pontificia Universidad Catolica de Chile), and M. Yampolsky (University of Toronto).