Integral points on hyperelliptic curves

Let C : Y-2 = a(n)X(n)+...+a(0) be a hyperelliptic curve with the ai rational integers, n >= 5, and the polynomial on the right- hand side irreducible. Let J be its Jacobian. We give a completely explicit upper bound for the integral points on the model C, provided we know at least one rational point on C and a Mordell Weil basis for J (Q). We also explain a powerful refinement of the Mordell - Weil sieve which, combined with the upper bound, is capable of determining all the integral points. Our method is illustrated by determining the integral points on the genus 2 hyperelliptic models Y-2 - Y = X-5 - X and (Y 2) = (X 5).