  Birational Yang-Baxter maps (`set-theoretical solutions of the Yang-Baxter
equation') are considered. A birational map $(x,y)\mapsto(u,v)$ is called
quadrirational, if its graph is also a graph of a birational map
$(x,v)\mapsto(u,y)$. We obtain a classification of quadrirational maps on
$\CP^1\times\CP^1$, and show that all of them satisfy the Yang-Baxter equation.
These maps possess a nice geometric interpretation in terms of linear pencil of
conics, the Yang-Baxter property being interpreted as a new incidence theorem
of the projective geometry of conics.
