  Since their introduction by Thurston, geodesic laminations on hyperbolic
surfaces occur in many contexts. In this paper, we propose a generalization of
geodesic laminations on locally CAT(0), complete, geodesic metric spaces, whose
boundary at infinity of the universal cover is endowed with a invariant total
cyclic order. Then we study these new objects on surfaces endowed with flat
structures and on finite metric graphs. The main result of the paper is a
theorem of classification of geodesic laminations on a compact surface endowed
with a flat structure. We also show that every finite metric graph, except
four, is the support of a geodesic lamination with uncountably many leaves none
of whose is eventually periodic.
