We give an overview of current computational methods for determining the rational
points on algebraic curves. We discuss how two methods, based on embedding a curve
in an abelian variety, provide a practical method for deciding whether the curve has
rational points and, if some additional technical condition is met, for the
determination of these points.
While we cannot prove the methods are always successful, we do have a heuristic
that makes us expect so. This means that the main problem becomes the
determination of rational points on abelian varieties, in particular the determination
of the free rank of the finitely generated group they form. We discuss some methods
that provide bounds on this rank.
Finally, we report on some recent progress on applying these methods to nonhyperelliptic
curves of genus .
