It is well known that the aberrations of optical systems due to such disturbances as chromatic variation may be treated by perturbation methods; it is also true that the calculation of geometrical aberrations may be interpreted as a perturbation problem. The perturbation concept is therefore adopted in the present paper which attempts to systematize and extend to higher orders the aberration theory of electron optics. Hamilton's 'equation of the characteristic function' is derived in the usual way from the variational equation for an unperturbed system. It is then assumed that the variational function, i.e. the integrand of the variational equation, depends on a 'perturbation parameter' and the corresponding perturbations of Hamilton's point characteristic function and of Hamilton's differential relation are obtained. The first-order terms, when separated, lead to the introduction of a pair of characteristic functions whose arguments may be-among other choices-the co-ordinates in the object and aperture surfaces; from these one may obtain the perturbation of the ray throughout its length. The perturbation of rays at a surface which is an image surface of the unperturbed system may be derived from a single characteristic function. Formulae simplify considerably if the unperturbed system is Gaussian or otherwise orthogonal. It is found that there exist two distinct pairs of second-order characteristic functions; this duality offers a means of checking second-order calculations. Once again, perturbations at an image surface may be evaluated from a single characteristic function. When calculations are taken beyond the first order, it is necessary to consider more than one parameter, since perturbation effects are then no longer additive. The theory is used to establish formulae for the geometrical, chromatic and mixed aberrations of two classes of systems; one contains all systems of rotational symmetry, the other all systems with curved axes. There are three appendices, the first giving certain transformations which may be applied to the perturbations of a variational function, the second discussing the use of complex co-ordinates, and the third giving a classification of electron-optical systems according to their aberration calculations. Although the theory is developed only to the second order, the extension to higher orders is straightforward.