I will explain elementary methods for computing the zeta function of an arithmetic scheme over a finite field, focusing on the case of hyperelliptic curves. It turns out that the number of points on such curves over a finite field can be packaged into traces of certain explicit matrices (called Hasse-Witt matrices) constructed from some coefficients of suitable powers of the defining polynomial. A highlight of this approach is that it is entirely elementary, in the sense that no cohomology is involved. I will also discuss efficient algorithms for computation and their complexity.