\indent I will discuss the problem of efficiently constructing curves of genus 1 and 2 over finite fields with a prescribed number N of points. In both cases, there are algorithms that, at least heuristically and in practice, run in time polynomial in log N. They are of complex analytic nature, using CM-techniques. Time permitting, I will also explain why these techniques are provably insufficient to efficiently construct genus-2 Jacobians with a presecribed number of points.