  The Hirsch Conjecture (1957) stated that the graph of a $d$-dimensional
polytope with $n$ facets cannot have (combinatorial) diameter greater than
$n-d$. That is, that any two vertices of the polytope can be connected by a
path of at most $n-d$ edges.
  This paper presents the first counterexample to the conjecture. Our polytope
has dimension 43 and 86 facets. It is obtained from a 5-dimensional polytope
with 48 facets which violates a certain generalization of the $d$-step
conjecture of Klee and Walkup.
