The failure behavior of geometrically asymmetric sandwich beams with a metal foam core is analytically and experimentally investigated. New initial failure modes of the asymmetric sandwich beams are observed under three-point bending, i.e., face yield, face wrinkling, core shear A, core shear AB, core shear A-AB, and indentation. It is shown that the initial failure modes of sandwich beams depend on the span of the beam, the thicknesses of top and bottom face sheets, core height and material properties. We derived the analytical formulae for the initial failure loads and then constructed the initial failure mechanism maps for the geometrically asymmetric sandwich beams. It is shown that the analytically predicted initial failure mechanism maps are in good agreement with the experimental results, which are clearly different from the symmetric sandwich beams. As a preliminary application, the minimum weight designs are presented for asymmetric metal sandwich beams.