In this talk, I will talk about one approach to study the Diophantine equation f(x)=g(y), which combines the tools from Galois theory, algebraic geometry and group theory.
In particular, I will explain how the methods are used in the joint work with Mike Zieve on the equation ax^m+bx^n+c=dy^p+ey^q.

The ideas and methods above are also used in a recent theorem
of Carney-Hortsch-Zieve, which says that for any polynomial f(x) in Q[x], the map f: Q -> Q, a -> f(a) is at most 6-to-1 off a finite subset of Q. I will state a much more general conjecture on uniform boundedness of rational preimages of rational functions on number fields, of which a quite special case implies the theorems of Mazur and Merel on rational torsion points of elliptic curves.