ON THE EQUATIONS Z(M)=F(X,Y) AND AX(P)+BY(Q)=CZ(R)

We investigate integer solutions of the superelliptic equation z(m) = F(x, y), (1) where F is a homogeneous polynomial with integer coeffici ents, and of the generalized Fermat equation Ax(p) + By(q) = Cz(r), (2 ) where A, B and C are non-zero integers. Call an integer solution (x, y,z) to such an equation proper if gcd(x, y, z)= 1. Using Faltings' Th eorem, we shall give criteria for these equations to have only finitel y many proper solutions. We examine (1) using a descent technique of K ummer, which allows us to obtain from any infinite set of proper solut ions to (1), infinitely many rational points on a curve of (usually) h igh genus, thus contradicting Faltings' Theorem (for example, this wor ks if F(t, 1) = 0 has three simple roots and m greater than or equal t o 4). We study (2) via a descent method which uses unramified covering s of P-1 \ {0, 1,infinity} of signature (p, q, r), and show that (2) h as only finitely many proper solutions if 1/p + 1/q + 1/r < 1. In case s where these coverings arise from modular curves, our descent leads n aturally to the approach of Hellegouarch and Frey to Fermat's Last The orem. We explain how their idea may be exploited for other examples of (2). We then collect together a variety of results for (2) when 1/p 1/q + 1/r greater than or equal to 1. In particular, we consider 'loc al-global' principles for proper solutions, and consider solutions in function fields.