In the 1930s, Emil Artin put forward an influential chain of conjectures
that concern the existence of p-adic points on projective hypersurfaces.
The status of these conjectures is uncertain : in some ways, the
conjecture is completely wrong, in others it is almost true.
The lecture will formulate Artin's conjectures in elementary terms and
then explain their relevance to diophantine analysis. The state of the
art will be summarized, with combinatorial methods at the focus of
attention.