The effect of eccentricity on the complex resonances of a conducting sphere eccentrically coated by a dielectric one is investigated. The solution of the boundary value problem is obtained in terms of spherical eigenwaves in combination with the translational addition theorem. The resonant wavenumbers are calculated from the numerical solution of a characteristic determinantal equation. The analysis is also specialized to the case of small eccentricity and closed form expressions are derived for the resonant wavenumbers. It is found that the restriction of small eccentricity is not severe. It is also found that under the influence of eccentricity the resonant wavenumbers are considerably displaced in the complex plane. This may be used in the identification of the position and the size of inhomogeneities inside spherical bodies.