  The base space of a semiuniversal unfolding of a hypersurface singularity
carries a rich geometry. By work of K. Saito and M. Saito is can be equipped
with the structure of a Frobenius manifold. By work of Cecotti and Vafa it can
be equipped with tt* geometry if the singularity is quasihomogeneous. tt*
geometry generalizes the notion of variation of Hodge structures.
  In the second part of this paper (chapters 6-8) Frobenius manifolds and tt*
geometry are constructed for any hypersurface singularity, using essentially
oscillating integrals; and the intimate relationship between polarized mixed
Hodge structures and this tt* geometry is worked out.
  In the first part (chapters 2-5) tt* geometry and Frobenius manifolds and
their relations are studied in general. To both of them flat connections with
poles are associated, with distinctive common and different properties. A frame
for a simultaneous construction is given.
