  We investigate semiconjugate rational functions, that is rational functions
$A,$ $B$ related by the functional equation $A\circ X=X\circ B$, where $X$ is a
rational function of degree at least two. We show that if $A$ and $B$ is a pair
of such functions, then either $B$ can be obtained from $A$ by a certain
iterative process, or $A$ and $B$ can be described in terms of orbifolds of
non-negative Euler characteristic on the Riemann sphere.
