For any finite field {\small $k=\fq$}, we explicitly describe the k-isogeny
classes of abelian surfaces defined over k and their behavior under
finite field extension. In particular, we determine the absolutely
simple abelian surfaces. Then, we analyze numerically what surfaces
are k-isogenous to the Jacobian of a smooth projective curve of
genus 2 defined over k. We prove some partial results suggested by
these numerical data. For instance, we show that every absolutely
simple abelian surface is k-isogenous to a Jacobian. Other facts
suggested by these numerical computations are that the polynomials
{\small $t^4+(1-2q)t^2+q^2$} (for all q) and {\small $t^4+(2-2q)t^2+q^2$} (for q
odd) are never the characteristic polynomial of Frobenius of a
Jacobian. These statements have been proved by E. Howe. The proof for the first polynomial is attached in an appendix.