[hansen]

The problems in the classical theory are easily understood. The charge and current densities of point particles are not smooth functions but distributions (think of the Dirac δ-“function”). If they act as the sources of the EM field the EM field itself becomes singular. Now if you try to solve the full Maxwell equations including the backreaction of matter & radiation fields you would have to multiply distributions which is ill defined.

There are similar problems in the quantum theory but the divergences are less severe and can be dealt with in a systematic way. Most physicist believe they will totally disappear in some more fundamental underlying theory. From a mathematicians point of view there is the hope that at least some QFTs are finite and the divergences are just an artifact of the construction & pertubation theory.