A mathematical model for scattering of electromagnetic waves from weakly nonlinear objects is developed. The constitutive relations are based on Volterra series, but additional, physically plausible, heuristic assumptions have to be introduced in order to solve the scattering problem. The general theory is discussed in connection with scattering from circular cylinders. These canonical problems demonstrate the new phenomena involved. It is shown that the first order effects of the nonlinear scattering problem involve modification of the linear scattering coefficients and production of new multipole terms at the fundamental frequency. In addition, part of the energy is transformed into harmonics. The corresponding problem of transient scattering is considered. The new effects of pole migration and pole creation are discussed. The present study contributes to understanding the theoretical aspects of nonlinear scattering, and may also provide a method for remote sensing of nonlinear targets.