Ultrafast electron microscopy in the space and time domains utilizes a pulsed electron probe to directly map structural dynamics of nanomaterials initiated by an optical pump pulse, in imaging, di raction, spectroscopy, and their combinations. It has demonstrated its capability in the studies of phase transitions, mechanical vibrations, and chemical reactions. Moreover, electrons can directly interact with photons via the near eld component of light scattering by nanostructures, and either gain or lose light quanta discretely in energy. By energetically selecting those electrons that exchanged photon energies, we can map this photon-electron interaction, and the technique is termed photon-induced near eld electron microscopy (PINEM). Here, we give an account of the theoretical understanding of PINEM. Experimentally, nanostructures such as a sphere, cylinder, strip, and triangle have been investigated. Theoretically, time-dependent Schrodinger and Dirac equations for an electron under light are directly solved to obtain analytical solutions. The interaction probability is expressed by the mechanical work done by an optical wave on a traveling electron, which can be evaluated analytically by the near eld components of the Rayleigh scattering for small spheres and thin cylinders, and numerically by the discrete dipole approximation for other geometries. Application in visualization of plasmon elds is discussed.