Resonance tongues in Hill's equations: a geometric approach Henk Broer, Dept. of Mathematics and Computing Science, University of Groningen, Blauwborgje 3, 9747 AC Groningen, The Netherlands Carles Sim\'o Dept. de Matem\`atica Aplicada i An\`alisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain Abstract The geometry of resonance tongues is considered in, mainly reversible, versions of Hill's equation, close to the classical Mathieu case. Hill's map assigns to each value of the multiparameter the corresponding Poincar\'e matrix. By an averaging method, the geometry of Hill's map locally can be understood in terms of cuspoid Whitney singularities. This adds robustness to the result. The algorithmic nature of the averaging method enables a pull--back to the resonance tongues of the original system.