By counting curves over finite fields, one obtains
a wealth of information about the cohomology of moduli spaces
of curves and of principally polarized abelian varieties.
Galois representations or mixed Hodge structures (or, in ideal
cases, motives) associated to modular forms make their appearance.
The modular forms are elliptic cusp forms for SL(2,Z) in genus 1,
Siegel cusp forms in genera 2 and 3, and, on the moduli space
M_3 of curves of genus 3, the first so-called Teichmüller modular
forms have recently been detected. This is a report on joint work
with Jonas Bergström and Gerard van der Geer.