During the first half of the talk I will give a survey about the Hitchin connection on the sheaf of non-abelian theta functions and its analogues in the theory of Conformal Blocks and TQFT. During the second part I will sketch the construction of the Hitchin connection (starting from the heat equation for abelian theta functions) from an algebro-geometric perspective. This is joint work with T. Baier, M. Bolognesi and J. Martens.