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The main result of this paper is that one cannot hear orientability of asurface with boundary. More precisely, we construct two isospectral flatsurfaces with boundary with the same Neumann spectrum, one orientable, theother non-orientable. For this purpose, we apply Sunada's and Buser's methodsin the framework of orbifolds. Choosing a symmetric tile in our construction,and adapting a folklore argument of Fefferman, we also show that the surfaceshave different Dirichlet spectra. These results were announced in the {\it C.R. Acad. Sci. Paris S\'er. I Math.}, volume 320 in 1995, but the full proofs sofar have only circulated in preprint form.

One can't hear orientability of surfaces