  We study the configurations of genus 2 curves on the Fano surfaces of cubic
threefolds. We establish a link between some involutive automorphisms acting on
such a surface S and genus 2 curves on S. We give a partial classification of
the Fano surfaces according to the automorphism group generated by these
involutions and determine the configurations of their genus 2 curves. We study
the Fano surface of the Klein cubic threefold for which the 55 genus 2 curves
generate a rank 25=h^{1,1} index 2 subgroup of the N\'eron-Severi group.
