Hill's equation with quasi--periodic forcing: resonance tongues, instability pockets and global phenomena 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 A simple example is considered of Hill's equation $\ddot{x}+(a^2+b\,p(t))x=0,$ where the forcing term $p,$ instead of periodic, is quasi--periodic with two frequencies. A geometric exploration is carried out of certain resonance tongues, containing instability pockets. This phenomenon in the perturbative case of small $|b|,$ can be explained by averaging. Next a numerical exploration is given for the global case of arbitrary $b,$ where some interesting phenomena occur. Regarding these, a detailed numerical investigation and tentative explanations are presented.