Abstract: I will talk about a computational problem inspired by the desire to improve the tables of curves over finite fields with many points (http://www.manypoints.org). Namely, if $q$ is a large prime power, how does one go about producing a genus-4 curve over $\mathbb F_q$ with many points? I will discuss the background to this problem and give a number of algorithms, one of which one expects (heuristically!) to produce a genus-4 curve whose number of points is quite close to the Weil upper bound in time $O\left(q^{3/4 + \epsilon}\right).$