Francisco Santos has announced a counterexample to the Hirsch conjecture:
“I will describe the construction of a 43-dimensional polytope with 86 facets and diameter bigger than 43. The proof is based on a generalization of the $d$-step theorem of Klee and Walkup.”
The above is from this website:
In July the result will presented at a conference in Seattle. More information is available at the above website.
The question of a polynomial upper bound is still open: http://gilkalai.wordpress.com/2010/05/10/francisco-santos-disproves-the-hirsch-conjecture/#comment-3040
“I am afraid my construction says nothing about the polynomiality. As far as I know $2n-2$ (for example) could still be an upper bound for the diameter.”
The question of polynomiality could still be the subject of Polymath3 as that question is open.
Francisco Santos has a website here:
Congratulations to him for solving this!