On the Equations zm = F(x, y) and Axp + Byq = Czr

We investigate integer solutions of the superelliptic equation where F is a homogeneous polynomialwith integer coefficients, and of the generalized Fermat equation where A, B and C are non-zero integers.Call an integer solution (x, y, z) to such an equation properif gcd(x, y, z) = 1. Using Faltings' Theorem, we shall givecriteria for these equations to have only finitely many propersolutions. We examine (1) using a descent technique of Kummer, which allowsus to obtain, from any infinite set of proper solutions to (1),infinitely many rational points on a curve of (usually) highgenus, thus contradicting Faltings' Theorem (for example, thisworks if F(t, 1) = 0 has three simple roots and m 4). We study (2) via a descent method which uses unramified coveringsof P₁ {0, 1, } of signature (p, q, r), and show that (2) hasonly finitely many proper solutions if l/p + l/q + 1/r <1. In cases where these coverings arise from modular curves,our descent leads naturally to the approach of Hellegouarchand Frey to Fermat's Last Theorem. We explain how their ideamay be exploited for other examples of (2). We then collect together a variety of results for (2) when 1/p+ 1/q + 1/r 1. In particular, we consider ‘local-global’principles for proper solutions, and consider solutions in functionfields.