TITLE Analytical and numerical detection of exponentially small phenomena} AUTHOR Carles Sim\'o Departament de Matem\`atica Aplicada i An\`alisi Universitat de Barcelona Gran Via de les Corts Catalanes, 585 08007 e-mail:carles@maia.ub.es ABSTRACT: Exponentially small phenomena occur in a wide variety of problems. Always dealing with problems which are analytical, we find these phenomena both in estimates of the remainders in normal forms of vector fields and diffeomorphisms and in averaging methods and, in general, in most of the phenomena which cannot be detected by using any fixed finite order approach of the classical perturbation theory. They also appear in phenomena like the delay of the bifurcation for fixed points and periodic solutions, in the case of slowly varying parameters, in adiabatic invariance and in the corresponding bifurcation diagrams of all these problems. They can be revealed by a suitable use of complex variable, by extending the phase space and time to suitable complex strips or by using a classical method letting the order to increase, up to some optimal value, when the small parameter tends to zero. Numerically they can be detected by a careful use of high accuracy arithmetics, leading in a natural way to the problem of optimization of all the algorithms involved.