Analytical expressions for the resonant frequencies in an electromagnetic cylindrical/spherical cavity with an off-axis inner electrically small dielectric sphere are derived, for both magnetic and electric modes. The walls of the cylindrical/spherical cavity are perfectly conducting. Cylindrical/spherical vector wave functions and related addition theorems, as well as expansion of cylindrical wave functions in terms of spherical ones, are used. Our results are useful in problems connected with resonant cavities containing small inhomogeneities, as well as in cases that we want to determine the permittivities or magnetic permeabilities of various materials, by measuring the resonant frequency shifts caused by the introduction of small samples of them, inside cavities. Graphical results for some of the lower order modes are given, for various values of the parameters and for both kinds of modes.