It would probably have been a big surprise for Hilbert to see that one of his famous problems was resolved with a negative answer:

As we know by the work of Davis, Putnam, Robinson and Matyasevitch, there is no algorithm that takes as input a multivariate polynomial f(x₁,…,x_r) over the integers and gives as output whether or not the the equation

This result establishes that number theory, and in particular the study of diophantine equations, is generally hard.

If instead of integer points on hypersurfaces, we consider rational points on curves, the picture changes dramatically. In the last six years, a suprisingly simple method, now commonly referred to as Mordell–Weil sieving, has been developed. A heuristic argument by Bjorn Poonen indicates that this method should always be able to decide if a projective curve has any rational points.

I will discuss the method and give some experimental evidence of its efficacy.