  The enumeration of points on (or off) the union of some linear or affine
subspaces over a finite field is dealt with in combinatorics via the
characteristic polynomial and in algebraic geometry via the zeta function. We
discuss the basic relations between these two points of view. Counting points
is also related to the $\ell$-adic cohomology of the arrangement (as a
variety). We describe the eigenvalues of the Frobenius map acting on this
cohomology, which corresponds to a finer decomposition of the zeta function.
The $\ell$-adic cohomology groups and their decomposition into eigenspaces are
shown to be fully determined by combinatorial data. Finally, it is shown that
the zeta function is determined by the topology of the corresponding complex
variety in some important cases.
