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Abstract: Ambiguity theory was the name which Galois used when he wrote of his own theory. For him, this theory was by no means limited to the study of algebraic equations: in his last letter, he mentioned that he was meditating on possible applications of ambiguity theory to transcendental analysis. Klein later paved the way by trying, as he put it, "to blend Galois' with Riemann's ideas". After recalling some early steps in this direction which were taken, with inequal success, by Lie, Picard, Drach... , we shall survey some recent developments confirming Galois' insight (Ramis' wild fundamental group, Malgrange's Galois groupoid attached to analytic foliations,...) and outline some applications. We shall also indicate how the theory of motives introduced by Grothendieck in arithmetic geometry should provide a Galois theory for some transcendental numbers (periods).