The physicists Cecotti and Vafa considered in 1991 a generalization of Hodge structures which is related to Simpson's harmonic bundles. Mathematicians took this up under different names (TERP structure, integrable twistor structure, non-commutative Hodge structure). It consists of a holomorphic vector bundle on P^1C with a flat connection on C* with poles of order 2 at 0 and infinity, with a flat real structure and a certain flat pairing on the bundle on C*. It arises via oscillating integrals in the case of Landau-Ginzburg models. Several results on Hodge structures have generalizations in this setting. Here Hodge structures and Stokes structures come together.