The first part of the talk is dedicated to a brief review, also from the historical side, of the strong ties between cubic hypersurfaces and rationality problems.
Then we will consider cubic hypersurfaces in the complex projective space of dimension 5 and their moduli.
As it is well known this is today a topical case where many conjectures are made about the locus of rational cubic hypersurfaces. Conjecturally this is union of loci of codimension one in the moduli space; these are related to moduli of K3 surfaces polarized by a line bundle of degree d = 2(n^2 + n + 1).
In the last part of the talk, I will describe the geometry of cubics corresponding to the case n = 3 and I will show how this is used to deduce that the universal K3 surface of degree d = 26 is rational (joint work with G. Farkas.)