  In this paper we give several conditions implying the irreducibility of the
algebraic curve P(x)-Q(y)=0, where P,Q are rational functions. We also apply
the results obtained to the functional equations P(f)=Q(g) and P(f)=cP(g),
where c\in C. For example, we show that for a generic pair of rational
functions P,Q the first equation has no non-constant solutions f,g meromorphic
on C whenever (\deg P-1)(\deg Q-1) \geq 2.
