**Title:** *Congruences for the number of rational points of varieties
defined over finite fields.
*

Smooth hypersurfaces of degree $d\le n$ are precisely Fano hypersurfaces. We will show how Bloch-Beilinson motivic philosophy, transposed to varieties over finite fields, allows to prove that all smooth Fano varieties carry a rational point over a finite field (Lang-Manin conjecture), thereby generalizing Chevalley-Warning-Ax-Katz' theorem. However, lacking a good motivic understanding in the singular case, we are not able to handle the case of singular Fano varieties. Yet we can nevertheless explain cohomologically Ax-Katz' theorem showing that the eigenvalues of Frobenius acting on \'etale cohomology have the required divisibility property (joint with N. Katz).