  For any given natural $d\ge 1$ we provide examples of rational self-maps of
complex projective plane $\pp^2$ of degree $d$ without (holomorphic) fixed
points. This makes a contrast with the situation in one dimension. We also
prove that the set of fixed point free rational self-maps of $\pp^2$ is closed
(modulo "degenerate" maps) in some natural topology on the space of rational
self-maps of $\pp^2$.
