The focusing of high current electron beams by means of several different systems of periodic electric and magnetic fields has been studied theoretically. One of these periodic electron optical systems studied, i.e. a periodic electric field produced by a series of annular disks held at alternately higher and lower potentials, has also been investigated experimentally in an electrolytic tank.It is found that if the distance between the disks or half period of the field is long (i.e. two or three times longer than the diameter of the apertures at the center of the disks), the potential on the axis may be expressed by a saw-toothed function. Under this condition, the equation for describing the electron beam profile may be solved analytically. The conditions which must be satisfied to give periodic focusing have been obtained.However, if the distance between the disks is short and comparable with the diameter of the aperture, the potential on the axis should be expressed by a cosine function. Under such a condition, the equation for describing the electron beam profile becomes a complicated non-linear differential equation. This non-linear differential equation has been treated in earlier research and was able to obtain its first order approximate solution. In order to get higher order approximations, a perturbation method may be used.This perturbation method has also been applied to treat other periodic electric and magnetic fields, such as axially symmetric magnetic fields, quadrupole electric and magnetic fields. Higher order periodic approximate solutions, as well as the required conditions to give periodic focusing, have been obtained for these fields. It can be shown by numerical examples that for periodic focusing of high current electron beams higher order approximation should be included.