We give a new and efficient method of sieving for rational points
on hyperelliptic curves. This method is often successful in proving that a
given hyperelliptic curve, suspected to have no rational points, does in fact
have no rational points; we have often found this to be the case even when our
curve has points over all localizations Qp. We illustrate the practicality of the
method with some examples of hyperelliptic curves of genus 1