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In this paper we give several conditions implying the irreducibility of thealgebraic curve P(x)-Q(y)=0, where P,Q are rational functions. We also applythe 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 rationalfunctions P,Q the first equation has no non-constant solutions f,g meromorphicon C whenever (\deg P-1)(\deg Q-1) \geq 2.

Algebraic curves P(x)-Q(y)=0 and functional equations