The nature of magnetic monopoles in an SO(3) gauge theory is explored by examining how they scatter charged particles. Peripheral scattering, where the momentum transfer is much less than the mass of the charged vector bosons, is just as it is in the earlier theory of Dirac. The scattering wave function is discussed with some rigor since its behavior is not uniform in the forward direction. The correspondence between the classical and quantum-mechanical scattering is displayed explicitly by relating scattering by the noncentral monopole force to the central force scattering by an attractive $\frac{1}{{r}^{2}}$ potential, for both the quantum-mechanical and classical systems. For deep scattering, the non-Abelian monopole exhibits features absent in the Dirac theory. The electromagnetic scattering departs from the result of the Dirac theory and charge-exchange processes occur, exciting the monopole into an electrically charged state. This process, which corresponds to the weak interaction of the unified SO(3) theory, is calculated in the distorted-wave Born approximation. The relation between the monopole in the non-Abelian gauge theory and that in Dirac's theory is investigated by carefully regulating the gauge transformation which connects them. This resolves some seeming paradoxes in the connection between the two theories.